Coordinates and Differential Elements
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[{"Name":"Coordinates and Differential Elements","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Coordinates","Duration":"16m 33s","ChapterTopicVideoID":8912,"CourseChapterTopicPlaylistID":144734,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8912.jpeg","UploadDate":"2021-11-18T08:37:03.1200000","DurationForVideoObject":"PT16M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.555","Text":"Hello. In this video, we\u0027re going to talk about coordinates."},{"Start":"00:03.555 ","End":"00:07.785","Text":"In order to describe the position of an object in space,"},{"Start":"00:07.785 ","End":"00:09.705","Text":"we need 3 data points."},{"Start":"00:09.705 ","End":"00:12.210","Text":"There are multiple kinds of coordinate systems, but in physics,"},{"Start":"00:12.210 ","End":"00:16.425","Text":"we tend to use 1 of 3 systems for any given problem."},{"Start":"00:16.425 ","End":"00:21.630","Text":"The 3 most common systems are Cartesian coordinates, x, y,"},{"Start":"00:21.630 ","End":"00:27.948","Text":"and z, polar coordinates and spherical coordinates."},{"Start":"00:27.948 ","End":"00:30.980","Text":"What I\u0027d like to do is look at these 3 types,"},{"Start":"00:30.980 ","End":"00:34.160","Text":"see some of the elements of them,"},{"Start":"00:34.160 ","End":"00:37.265","Text":"and talk about how we can transition from one to the other mathematically,"},{"Start":"00:37.265 ","End":"00:39.935","Text":"when looking to describe a given point."},{"Start":"00:39.935 ","End":"00:43.700","Text":"The first coordinates we\u0027re going to talk about are Cartesian coordinates."},{"Start":"00:43.700 ","End":"00:47.291","Text":"You might be familiar with them from prior class you\u0027ve taken particularly in math."},{"Start":"00:47.291 ","End":"00:50.390","Text":"We describe a point with values of x, y,"},{"Start":"00:50.390 ","End":"00:54.220","Text":"or z, and use these variables to locate our points in space."},{"Start":"00:54.220 ","End":"00:57.267","Text":"If we want to describe this red point, we just circled,"},{"Start":"00:57.267 ","End":"00:59.900","Text":"what we can do is find the values in x, y,"},{"Start":"00:59.900 ","End":"01:02.190","Text":"and z using orthogonal lines,"},{"Start":"01:02.190 ","End":"01:04.200","Text":"that is lines that have right angles."},{"Start":"01:04.200 ","End":"01:06.140","Text":"We draw from the red point."},{"Start":"01:06.140 ","End":"01:09.560","Text":"We can draw an orthogonal line to the z-axis,"},{"Start":"01:09.560 ","End":"01:14.420","Text":"and that will give us our value in z in terms of the height by measuring that down,"},{"Start":"01:14.420 ","End":"01:18.260","Text":"and then we draw a line to the x, y plane."},{"Start":"01:18.260 ","End":"01:19.910","Text":"From the point where it comes out in x, y plane,"},{"Start":"01:19.910 ","End":"01:23.240","Text":"we draw one orthogonal line to the x-axis and another one to the y-axis,"},{"Start":"01:23.240 ","End":"01:27.020","Text":"and we can use these 3 lines to measure the values of x,"},{"Start":"01:27.020 ","End":"01:29.155","Text":"y, and z and locate our point."},{"Start":"01:29.155 ","End":"01:34.625","Text":"Now let\u0027s describe the same point using cylindrical or polar coordinates."},{"Start":"01:34.625 ","End":"01:36.830","Text":"We\u0027re going to look instead of for x, y,"},{"Start":"01:36.830 ","End":"01:39.200","Text":"and z for r Theta and z."},{"Start":"01:39.200 ","End":"01:40.820","Text":"We can start with z. It\u0027s rather simple."},{"Start":"01:40.820 ","End":"01:41.960","Text":"It\u0027s the same z from before."},{"Start":"01:41.960 ","End":"01:44.300","Text":"It\u0027s the height rising from the plane."},{"Start":"01:44.300 ","End":"01:46.555","Text":"We can find it in the same way,"},{"Start":"01:46.555 ","End":"01:48.170","Text":"and if we want to find r,"},{"Start":"01:48.170 ","End":"01:51.350","Text":"what we\u0027re going to do is draw that same orthogonal or"},{"Start":"01:51.350 ","End":"01:54.995","Text":"perpendicular line from the red point to the z-axis,"},{"Start":"01:54.995 ","End":"02:01.010","Text":"and that distance is r. r is the distance that the point is from the z-axis."},{"Start":"02:01.010 ","End":"02:03.605","Text":"Now, to find the Theta angle,"},{"Start":"02:03.605 ","End":"02:05.210","Text":"what we\u0027re going to do is from the red point,"},{"Start":"02:05.210 ","End":"02:08.925","Text":"we\u0027re going to drop down a line like we did before to the plane."},{"Start":"02:08.925 ","End":"02:12.560","Text":"The angle between the r line along this plane,"},{"Start":"02:12.560 ","End":"02:15.440","Text":"the same r line you drew the z-axis before."},{"Start":"02:15.440 ","End":"02:20.710","Text":"The angle between that and the x-axis from before is your Theta angle."},{"Start":"02:20.710 ","End":"02:22.400","Text":"When we find the angle of Theta,"},{"Start":"02:22.400 ","End":"02:30.815","Text":"we can describe the angle of this object in relationship to the x-axis."},{"Start":"02:30.815 ","End":"02:34.790","Text":"If you\u0027re given a point described in Cartesian coordinates,"},{"Start":"02:34.790 ","End":"02:37.145","Text":"you can easily transition into cylindrical,"},{"Start":"02:37.145 ","End":"02:40.340","Text":"and if you\u0027re given a cylindrical coordinate object,"},{"Start":"02:40.340 ","End":"02:44.420","Text":"you can easily transpose it into Cartesian coordinate form."},{"Start":"02:44.420 ","End":"02:46.940","Text":"Basically what you need to do is if you\u0027re given,"},{"Start":"02:46.940 ","End":"02:49.280","Text":"let\u0027s say something described in r,"},{"Start":"02:49.280 ","End":"02:52.340","Text":"Theta, and z, you need to transpose that into x, y, and z."},{"Start":"02:52.340 ","End":"02:54.980","Text":"For example, if you\u0027re to do that here,"},{"Start":"02:54.980 ","End":"02:57.775","Text":"the first thing we knew is z is the same z,"},{"Start":"02:57.775 ","End":"03:00.590","Text":"so we can stick with that, and to find x,"},{"Start":"03:00.590 ","End":"03:05.765","Text":"what we do is take our r value and multiply it by cosine of Theta."},{"Start":"03:05.765 ","End":"03:10.330","Text":"The way we derive this is if we transpose the point down to the x,"},{"Start":"03:10.330 ","End":"03:14.495","Text":"y plane, we can draw a line from that point to the x-axis,"},{"Start":"03:14.495 ","End":"03:15.740","Text":"which is going to be at a right angle,"},{"Start":"03:15.740 ","End":"03:19.040","Text":"and that way we know that x over r equals cosine Theta,"},{"Start":"03:19.040 ","End":"03:20.450","Text":"so if we multiply that out,"},{"Start":"03:20.450 ","End":"03:23.525","Text":"we end up with x equaling r times cosine Theta."},{"Start":"03:23.525 ","End":"03:26.120","Text":"To find the y-value, we do a very similar thing,"},{"Start":"03:26.120 ","End":"03:27.965","Text":"y is equal to r sine Theta,"},{"Start":"03:27.965 ","End":"03:31.340","Text":"and the reason for that is because the angle Theta as I just drew there,"},{"Start":"03:31.340 ","End":"03:35.360","Text":"is between r and the right angle towards the y-axis,"},{"Start":"03:35.360 ","End":"03:38.300","Text":"so y over r equals sine Theta,"},{"Start":"03:38.300 ","End":"03:41.280","Text":"and therefore y equals r sine Theta."},{"Start":"03:41.280 ","End":"03:43.855","Text":"Now, if you want to do things the other way around,"},{"Start":"03:43.855 ","End":"03:45.985","Text":"that\u0027s also rather simple."},{"Start":"03:45.985 ","End":"03:50.800","Text":"The easiest way to find r is to think of it in terms of Pythagorean theorem."},{"Start":"03:50.800 ","End":"03:54.960","Text":"r is equal to the square root of x^2 plus y^2"},{"Start":"03:54.960 ","End":"03:59.260","Text":"because if you take that y-axis line and move it forward a little bit,"},{"Start":"03:59.260 ","End":"04:02.185","Text":"you can see it makes a perfect right-angle triangle."},{"Start":"04:02.185 ","End":"04:04.540","Text":"If we\u0027re trying to find the angle of Theta,"},{"Start":"04:04.540 ","End":"04:06.678","Text":"we use tangent of Theta equals y over x."},{"Start":"04:06.678 ","End":"04:08.815","Text":"Because again, with that same triangle,"},{"Start":"04:08.815 ","End":"04:14.515","Text":"you can use the geometrical formulas to find that that is the tangent angle."},{"Start":"04:14.515 ","End":"04:17.170","Text":"You could do it with the sine using r,"},{"Start":"04:17.170 ","End":"04:23.937","Text":"but if you see x as the side next to the angle and y is the side across from it,"},{"Start":"04:23.937 ","End":"04:26.105","Text":"you can use the tangent, it\u0027s rather simple."},{"Start":"04:26.105 ","End":"04:29.360","Text":"Now the last important thing is just realize what values r and"},{"Start":"04:29.360 ","End":"04:32.780","Text":"Theta can take on in a cylindrical coordinate graph."},{"Start":"04:32.780 ","End":"04:37.055","Text":"What we see here is that r can only be positive,"},{"Start":"04:37.055 ","End":"04:38.615","Text":"r can be from 0 to infinity."},{"Start":"04:38.615 ","End":"04:41.630","Text":"The reason for that is we\u0027re going to use the Theta angle to"},{"Start":"04:41.630 ","End":"04:45.710","Text":"position it around the z-axis or around the origin."},{"Start":"04:45.710 ","End":"04:47.960","Text":"We don\u0027t need it to take on negative values because"},{"Start":"04:47.960 ","End":"04:50.555","Text":"its data can go from anywhere from 0 to 2 Pi,"},{"Start":"04:50.555 ","End":"04:54.135","Text":"meaning we can move it entirely around the axis in a full circle."},{"Start":"04:54.135 ","End":"04:56.705","Text":"That line that I drew there in blue,"},{"Start":"04:56.705 ","End":"04:58.490","Text":"instead of describing with a negative r,"},{"Start":"04:58.490 ","End":"04:59.970","Text":"you can describe with a positive r,"},{"Start":"04:59.970 ","End":"05:05.555","Text":"and the Theta would be something like maybe 1.6 Pi, 1.7 Pi."},{"Start":"05:05.555 ","End":"05:11.420","Text":"You could make it go 290 maybe that\u0027s 300 degrees around the circle to describe that,"},{"Start":"05:11.420 ","End":"05:13.820","Text":"instead of using a negative r. In general,"},{"Start":"05:13.820 ","End":"05:16.850","Text":"we could describe it as a negative angle of negative 60,"},{"Start":"05:16.850 ","End":"05:20.180","Text":"negative 80, but we don\u0027t need to deal with that right now."},{"Start":"05:20.180 ","End":"05:24.575","Text":"The last step is describing things in terms of a spherical coordinate system."},{"Start":"05:24.575 ","End":"05:26.844","Text":"We\u0027re going to describe things in terms of r,"},{"Start":"05:26.844 ","End":"05:28.940","Text":"Theta, and Phi."},{"Start":"05:28.940 ","End":"05:31.520","Text":"Now, it\u0027s important to know that the difference between"},{"Start":"05:31.520 ","End":"05:37.020","Text":"cylindrical and spherical graphing is that the r is not the same thing."},{"Start":"05:37.020 ","End":"05:40.460","Text":"In spherical graphing, we tend to keep it as a symbol r,"},{"Start":"05:40.460 ","End":"05:44.855","Text":"people tend to describe the r in cylindrical graphing as row."},{"Start":"05:44.855 ","End":"05:48.545","Text":"The reason that we\u0027re using the letter r, the symbol r,"},{"Start":"05:48.545 ","End":"05:53.750","Text":"for our class is that oftentimes in physics we use the symbol Rho to describe density."},{"Start":"05:53.750 ","End":"05:56.180","Text":"You just need to know that we\u0027re not talking about the same thing."},{"Start":"05:56.180 ","End":"05:57.995","Text":"You\u0027ll see why in a second."},{"Start":"05:57.995 ","End":"06:02.795","Text":"With that said, let\u0027s turn our attention for a minute to spherical coordinates."},{"Start":"06:02.795 ","End":"06:07.190","Text":"The r is different because instead of being measured from the z-axis,"},{"Start":"06:07.190 ","End":"06:08.795","Text":"the r is measured from the origin."},{"Start":"06:08.795 ","End":"06:12.965","Text":"Again, the r is measuring a distance from the origin and not from the z-axis."},{"Start":"06:12.965 ","End":"06:14.480","Text":"This is the main difference here."},{"Start":"06:14.480 ","End":"06:19.600","Text":"Now we use that r value to find our Phi value."},{"Start":"06:19.600 ","End":"06:25.080","Text":"Phi is the angle from the r line that is there to the z-axis."},{"Start":"06:25.080 ","End":"06:27.675","Text":"Now remember we\u0027re describing the same point,"},{"Start":"06:27.675 ","End":"06:30.660","Text":"and if we drop that point down to the x,"},{"Start":"06:30.660 ","End":"06:31.940","Text":"y plane like we did before,"},{"Start":"06:31.940 ","End":"06:34.790","Text":"transpose it there, we can find the same Theta angle as before."},{"Start":"06:34.790 ","End":"06:38.000","Text":"Theta is the same as it was before."},{"Start":"06:38.000 ","End":"06:46.445","Text":"One last note is that the line between the origin and the transposed object point,"},{"Start":"06:46.445 ","End":"06:49.003","Text":"we\u0027re going to call that r_xy,"},{"Start":"06:49.003 ","End":"06:50.960","Text":"for r drop down to the x, y plane."},{"Start":"06:50.960 ","End":"06:52.400","Text":"It\u0027s not always called that,"},{"Start":"06:52.400 ","End":"06:55.015","Text":"but it\u0027s really useful for some of the functions as we\u0027ll see below."},{"Start":"06:55.015 ","End":"06:57.410","Text":"Of course, because this whole thing is a rectangle,"},{"Start":"06:57.410 ","End":"07:03.245","Text":"the same line that\u0027s parallel to it above is also r_xy."},{"Start":"07:03.245 ","End":"07:05.810","Text":"Again, measuring that length from the z-axis that we would"},{"Start":"07:05.810 ","End":"07:08.870","Text":"normally in cylindrical graphing measure as r,"},{"Start":"07:08.870 ","End":"07:12.650","Text":"we\u0027re going to call here our r_xy and it\u0027s going to be important for us."},{"Start":"07:12.650 ","End":"07:15.710","Text":"In order to find the relationships between"},{"Start":"07:15.710 ","End":"07:21.215","Text":"the spherical and the Cartesian and cylindrical coordinates,"},{"Start":"07:21.215 ","End":"07:24.020","Text":"we\u0027re going to use those things that are equal between some of them."},{"Start":"07:24.020 ","End":"07:27.080","Text":"Between cylindrical and spherical coordinates,"},{"Start":"07:27.080 ","End":"07:28.340","Text":"we know that Theta is the same,"},{"Start":"07:28.340 ","End":"07:29.690","Text":"we know that r_xy from"},{"Start":"07:29.690 ","End":"07:32.645","Text":"spherical coordinates is the same as r from cylindrical coordinates."},{"Start":"07:32.645 ","End":"07:34.580","Text":"When we want to find similar things,"},{"Start":"07:34.580 ","End":"07:36.605","Text":"we\u0027re going to use those terms a lot."},{"Start":"07:36.605 ","End":"07:39.410","Text":"For example, if we\u0027re going to Cartesian coordinates,"},{"Start":"07:39.410 ","End":"07:42.155","Text":"if we want to find the value of z,"},{"Start":"07:42.155 ","End":"07:45.610","Text":"we know that it\u0027s on a right angle with the line r_xy,"},{"Start":"07:45.610 ","End":"07:48.715","Text":"and we can use the angle Phi to find them."},{"Start":"07:48.715 ","End":"07:51.875","Text":"Of course, we know Phi, because it\u0027s given in our spherical coordinates."},{"Start":"07:51.875 ","End":"07:57.590","Text":"We know that z over r is going to equal the cosine of Phi because again,"},{"Start":"07:57.590 ","End":"08:03.875","Text":"you can see the angle Phi is adjacent to r and adjacent to z."},{"Start":"08:03.875 ","End":"08:08.610","Text":"Now if we multiply that out, we get z equals r cosine Phi."},{"Start":"08:08.610 ","End":"08:12.175","Text":"Now that we have z, we should go and find x and y, of course."},{"Start":"08:12.175 ","End":"08:15.970","Text":"Where we\u0027re going to do that is find the value first of r_xy and we\u0027re going"},{"Start":"08:15.970 ","End":"08:19.495","Text":"to use the exact same way we use r in our cylindrical format earlier."},{"Start":"08:19.495 ","End":"08:23.890","Text":"To find r_xy we\u0027re going to do a very similar operation that we just did with z."},{"Start":"08:23.890 ","End":"08:29.035","Text":"We know that r_xy is across from the angle say Phi."},{"Start":"08:29.035 ","End":"08:36.205","Text":"We know that r_xy over r equals sine of Phi."},{"Start":"08:36.205 ","End":"08:40.435","Text":"If we multiply that out, we get r_xy equals r sine of Phi."},{"Start":"08:40.435 ","End":"08:44.815","Text":"We can do is transpose that r_xy down below onto the x-y plane,"},{"Start":"08:44.815 ","End":"08:46.615","Text":"which is where the name comes from again."},{"Start":"08:46.615 ","End":"08:51.580","Text":"We can do the exact same functions that we did before in our cylindrical coordinates."},{"Start":"08:51.580 ","End":"08:56.020","Text":"We know that x as it was before equal to r cosine of Theta."},{"Start":"08:56.020 ","End":"08:59.350","Text":"Now we know that it\u0027s equal to r_xy cosine of Theta."},{"Start":"08:59.350 ","End":"09:02.830","Text":"If we take that and extrapolate it to the next step,"},{"Start":"09:02.830 ","End":"09:09.520","Text":"we know that r_xy is actually equal to r sine of Phi,"},{"Start":"09:09.520 ","End":"09:13.855","Text":"so r sine of Phi times cosine of Theta equals x."},{"Start":"09:13.855 ","End":"09:18.111","Text":"Similarly for the y, we do the same thing that we did before,"},{"Start":"09:18.111 ","End":"09:20.470","Text":"r sine of Theta equal to y."},{"Start":"09:20.470 ","End":"09:25.300","Text":"Now r_xy sine of Theta equals y and extrapolating that to the next step,"},{"Start":"09:25.300 ","End":"09:29.485","Text":"r sine of Phi times r sine of Theta equals y,"},{"Start":"09:29.485 ","End":"09:33.865","Text":"giving us both x and y in terms of our polar coordinates."},{"Start":"09:33.865 ","End":"09:36.655","Text":"Now that you can take spherical coordinates"},{"Start":"09:36.655 ","End":"09:38.680","Text":"and transpose them into Cartesian coordinates,"},{"Start":"09:38.680 ","End":"09:40.618","Text":"it\u0027s useful to know how to do things the other way around,"},{"Start":"09:40.618 ","End":"09:42.700","Text":"so you really can describe things any way you"},{"Start":"09:42.700 ","End":"09:45.400","Text":"want and any way you need in terms of a problem you\u0027re given."},{"Start":"09:45.400 ","End":"09:48.130","Text":"You need to know how to take something given to you in x, y,"},{"Start":"09:48.130 ","End":"09:52.345","Text":"and z terms and describe it in terms of r, Theta, and Phi."},{"Start":"09:52.345 ","End":"09:55.285","Text":"The first thing we can start with this r. We know that"},{"Start":"09:55.285 ","End":"09:58.720","Text":"r is the hypotenuse of the triangle zr,"},{"Start":"09:58.720 ","End":"10:00.820","Text":"r_xy, r_xy again,"},{"Start":"10:00.820 ","End":"10:03.040","Text":"is our values are from the cylindrical format."},{"Start":"10:03.040 ","End":"10:10.105","Text":"The way that we find the value of r in spherical terms is we do a Pythagorean theorem."},{"Start":"10:10.105 ","End":"10:17.380","Text":"We take r as equal to the square root of r_xy^2 and z^2."},{"Start":"10:17.380 ","End":"10:22.315","Text":"Now if you recall from the cylindrical coordinates,"},{"Start":"10:22.315 ","End":"10:27.835","Text":"r_xy is actually the hypotenuse of the triangle xy, r_xy."},{"Start":"10:27.835 ","End":"10:33.340","Text":"We can describe that. Instead of as r_xy^2 as x^2 plus y^2."},{"Start":"10:33.340 ","End":"10:36.355","Text":"We can extrapolate that out as r,"},{"Start":"10:36.355 ","End":"10:38.095","Text":"in our spherical format,"},{"Start":"10:38.095 ","End":"10:40.600","Text":"equaling the square root of x,"},{"Start":"10:40.600 ","End":"10:42.510","Text":"where x^2 plus y^2,"},{"Start":"10:42.510 ","End":"10:47.250","Text":"which is the same as r_xy^2 plus z^2."},{"Start":"10:47.250 ","End":"10:50.115","Text":"In the end, what we come out to simply put is r"},{"Start":"10:50.115 ","End":"10:53.975","Text":"equals the square root of x^2 plus y^2 plus z^2."},{"Start":"10:53.975 ","End":"10:55.990","Text":"Now to further prove to you that the r in"},{"Start":"10:55.990 ","End":"10:58.968","Text":"spherical format and the r in cylindrical format are not the same,"},{"Start":"10:58.968 ","End":"11:03.100","Text":"you can see that they\u0027re equal to different things in terms of Cartesian coordinates."},{"Start":"11:03.100 ","End":"11:08.320","Text":"The r in cylindrical format is equal to the square root of x^2 plus y^2 and"},{"Start":"11:08.320 ","End":"11:13.690","Text":"the r in spherical format is equal to square root of x^2 plus y^2 plus z^2."},{"Start":"11:13.690 ","End":"11:16.300","Text":"Now, the Theta stays the same,"},{"Start":"11:16.300 ","End":"11:20.650","Text":"so it\u0027s easy to find, tangent of Theta is still equal to y over x, that hasn\u0027t changed."},{"Start":"11:20.650 ","End":"11:24.400","Text":"But to find Phi, we\u0027re going to actually use the r that we just found."},{"Start":"11:24.400 ","End":"11:29.635","Text":"We know that we can use the cosine of Phi to find it because"},{"Start":"11:29.635 ","End":"11:36.220","Text":"r and z are both adjacent to the Phi angle."},{"Start":"11:36.220 ","End":"11:39.280","Text":"We know that the cosine of Phi equals z"},{"Start":"11:39.280 ","End":"11:42.490","Text":"over r or more accurately z over r equals cosine Phi."},{"Start":"11:42.490 ","End":"11:45.415","Text":"We can simplify that or really extrapolate that out by"},{"Start":"11:45.415 ","End":"11:48.505","Text":"making r in terms of Cartesian coordinates that we found above."},{"Start":"11:48.505 ","End":"11:50.155","Text":"Instead, it\u0027s being z over r,"},{"Start":"11:50.155 ","End":"11:52.810","Text":"we\u0027re going to say cosine of Phi is equal to z"},{"Start":"11:52.810 ","End":"11:56.500","Text":"over the square root of x^2 plus y^2 plus z^2."},{"Start":"11:56.500 ","End":"12:01.240","Text":"Now we can describe all of our spherical coordinates in terms of Cartesian values."},{"Start":"12:01.240 ","End":"12:05.410","Text":"The next thing to do is describe how these values work,"},{"Start":"12:05.410 ","End":"12:07.480","Text":"r is the same as the r from"},{"Start":"12:07.480 ","End":"12:11.440","Text":"the cylindrical coordinates in the sense that it can only be positive."},{"Start":"12:11.440 ","End":"12:13.060","Text":"It goes from 0 to infinity,"},{"Start":"12:13.060 ","End":"12:16.510","Text":"but it must be positive for a similar reason as before."},{"Start":"12:16.510 ","End":"12:18.760","Text":"Theta, again, is going to stay as it was,"},{"Start":"12:18.760 ","End":"12:20.050","Text":"it goes from 0 to 2 Pi."},{"Start":"12:20.050 ","End":"12:24.775","Text":"It really functions identically to the way it functions in a cylindrical coordinates."},{"Start":"12:24.775 ","End":"12:27.610","Text":"Phi goes only from 0 to Pi."},{"Start":"12:27.610 ","End":"12:29.350","Text":"Now I\u0027m sure that you\u0027re asking yourself,"},{"Start":"12:29.350 ","End":"12:33.730","Text":"why does Theta goes from 0 to 2Pi and Phi only goes from 0 to Pi?"},{"Start":"12:33.730 ","End":"12:35.320","Text":"I\u0027ll describe that in a second."},{"Start":"12:35.320 ","End":"12:37.930","Text":"That\u0027s our last subject for this lecture."},{"Start":"12:37.930 ","End":"12:41.860","Text":"We\u0027re faced with an issue here where we can describe things"},{"Start":"12:41.860 ","End":"12:45.100","Text":"in terms of Phi being from 0 to 2Pi."},{"Start":"12:45.100 ","End":"12:46.420","Text":"We could do that if we wanted,"},{"Start":"12:46.420 ","End":"12:51.880","Text":"but it would cause a problem for us in terms of a double representation."},{"Start":"12:51.880 ","End":"12:54.460","Text":"What I mean by that is the following."},{"Start":"12:54.460 ","End":"13:02.640","Text":"If we have Phi going all the way around to any point 0 to 2Pi,"},{"Start":"13:02.640 ","End":"13:06.330","Text":"we can describe the same point in terms of Phi being a value"},{"Start":"13:06.330 ","End":"13:10.245","Text":"between 1Pi and 2Pi or in terms of Theta being a value between 1Pi and 2Pi."},{"Start":"13:10.245 ","End":"13:12.600","Text":"I\u0027ll illustrate what I mean here with an example."},{"Start":"13:12.600 ","End":"13:16.650","Text":"Let\u0027s say we want to describe an object at this red point,"},{"Start":"13:16.650 ","End":"13:20.370","Text":"which can be described one way as taking"},{"Start":"13:20.370 ","End":"13:24.415","Text":"Phi and giving it a value of something like 1.7Pi."},{"Start":"13:24.415 ","End":"13:25.810","Text":"Let\u0027s say it\u0027s, I don\u0027t know,"},{"Start":"13:25.810 ","End":"13:28.240","Text":"30 degrees from the z-axis."},{"Start":"13:28.240 ","End":"13:31.705","Text":"That would be 330 degrees around."},{"Start":"13:31.705 ","End":"13:34.720","Text":"We could describe it that way and then have our r be positive"},{"Start":"13:34.720 ","End":"13:38.575","Text":"and have a minimal Theta value and be fine."},{"Start":"13:38.575 ","End":"13:41.830","Text":"Now the problem here is that we can also describe"},{"Start":"13:41.830 ","End":"13:49.360","Text":"the same point with keeping our Phi at 30 degrees instead of 330 degrees."},{"Start":"13:49.360 ","End":"13:53.560","Text":"If we just keep our Phi at 30 degrees,"},{"Start":"13:53.560 ","End":"13:58.060","Text":"we keep our r in the same place."},{"Start":"13:58.060 ","End":"14:00.805","Text":"We can move like that."},{"Start":"14:00.805 ","End":"14:10.345","Text":"When we give our Theta some value of around 270 degrees,"},{"Start":"14:10.345 ","End":"14:15.400","Text":"we bring our point to that same place"},{"Start":"14:15.400 ","End":"14:19.945","Text":"there without having to use a Phi value that\u0027s greater than 1Pi."},{"Start":"14:19.945 ","End":"14:22.270","Text":"Now, this seems like it would be an advantage."},{"Start":"14:22.270 ","End":"14:23.680","Text":"We can describe things in multiple ways, but actually,"},{"Start":"14:23.680 ","End":"14:28.405","Text":"it gives us a problem when we\u0027re trying to skim out integrals."},{"Start":"14:28.405 ","End":"14:33.940","Text":"An integral is essentially the sum of different locations, as you\u0027ll see later."},{"Start":"14:33.940 ","End":"14:37.915","Text":"When we do this, we\u0027re going to be counting each point twice."},{"Start":"14:37.915 ","End":"14:40.795","Text":"This can give us a lot of problems going forward."},{"Start":"14:40.795 ","End":"14:42.805","Text":"Just to illustrate this further,"},{"Start":"14:42.805 ","End":"14:49.420","Text":"you can see here how if our Phi is limited to being up to 1Pi and not greater,"},{"Start":"14:49.420 ","End":"14:51.160","Text":"we keep it at an angle of 30."},{"Start":"14:51.160 ","End":"14:55.240","Text":"We can have our Theta going round up to 2Pi so it can make a full circle."},{"Start":"14:55.240 ","End":"14:59.365","Text":"Now we\u0027re only counting that point once."},{"Start":"14:59.365 ","End":"15:03.743","Text":"Same is true here if we have a greater angle for Phi,"},{"Start":"15:03.743 ","End":"15:07.960","Text":"the same goes true as you get lower and lower and lower on our graph here."},{"Start":"15:07.960 ","End":"15:13.165","Text":"But if we could have Phi given a value greater than 1Pi,"},{"Start":"15:13.165 ","End":"15:20.035","Text":"say that value there where it\u0027s going more than halfway around the circle,"},{"Start":"15:20.035 ","End":"15:22.225","Text":"we\u0027re going to count these same points twice."},{"Start":"15:22.225 ","End":"15:23.290","Text":"This really is problematic,"},{"Start":"15:23.290 ","End":"15:27.950","Text":"we don\u0027t want to be over-counting our points when we\u0027re making an integral."},{"Start":"15:28.320 ","End":"15:32.470","Text":"Let it suffice to say for now that we\u0027re just going to have"},{"Start":"15:32.470 ","End":"15:37.240","Text":"Phi go from 0 to 1Pi and Theta go from 0 up to 2Pi,"},{"Start":"15:37.240 ","End":"15:42.535","Text":"meaning that Phi will be responsible for smaller angles up to 180 degrees,"},{"Start":"15:42.535 ","End":"15:47.800","Text":"whereas Theta can go up to 360 degrees and go all the way around our origin,"},{"Start":"15:47.800 ","End":"15:50.120","Text":"all the way around the z-axis."},{"Start":"15:50.340 ","End":"15:52.600","Text":"We\u0027ll leave it at that for now."},{"Start":"15:52.600 ","End":"15:55.360","Text":"If you remember that things will be simple and we can perform"},{"Start":"15:55.360 ","End":"15:59.305","Text":"any function we want within spherical coordinates."},{"Start":"15:59.305 ","End":"16:04.180","Text":"One last point to mention is that other lectures, other textbooks,"},{"Start":"16:04.180 ","End":"16:08.680","Text":"other people you come across may change Theta and Phi from time to time,"},{"Start":"16:08.680 ","End":"16:10.930","Text":"meaning that everything that we just wrote,"},{"Start":"16:10.930 ","End":"16:13.990","Text":"we would switch our Theta and our Phi in terms of their places."},{"Start":"16:13.990 ","End":"16:15.670","Text":"Phi would describe the angle on"},{"Start":"16:15.670 ","End":"16:21.820","Text":"the x-y plane and Theta would describe the angle between r and the z-axis."},{"Start":"16:21.820 ","End":"16:27.006","Text":"As a result, your Theta would only go from 0 to 1Pi and Phi would go from 0 to 2Pi,"},{"Start":"16:27.006 ","End":"16:33.290","Text":"and all the angles below and our formulas here would be switched. That ends our lecture."}],"ID":9186},{"Watched":false,"Name":"Differential Elements","Duration":"20m 16s","ChapterTopicVideoID":8913,"CourseChapterTopicPlaylistID":144734,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8913.jpeg","UploadDate":"2017-03-21T08:22:13.9800000","DurationForVideoObject":"PT20M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this lecture, we\u0027re going to continue our discussion of"},{"Start":"00:02.610 ","End":"00:05.910","Text":"coordinate systems by exploring differential elements."},{"Start":"00:05.910 ","End":"00:07.845","Text":"These are things that measure your length,"},{"Start":"00:07.845 ","End":"00:10.305","Text":"your area or surface area, and your volume."},{"Start":"00:10.305 ","End":"00:11.970","Text":"We can do that in all three coordinate systems,"},{"Start":"00:11.970 ","End":"00:13.290","Text":"and the formulas are a little different."},{"Start":"00:13.290 ","End":"00:15.420","Text":"If we start with Cartesian coordinates,"},{"Start":"00:15.420 ","End":"00:18.735","Text":"we\u0027re going to measure dl which is for lengths,"},{"Start":"00:18.735 ","End":"00:22.635","Text":"dS for surface area,"},{"Start":"00:22.635 ","End":"00:24.120","Text":"and dV for volume."},{"Start":"00:24.120 ","End":"00:25.380","Text":"Dl is for length."},{"Start":"00:25.380 ","End":"00:27.945","Text":"It\u0027s also symbolized sometimes by ds."},{"Start":"00:27.945 ","End":"00:33.855","Text":"That comes from a measure of distance on a path that we\u0027ll talk about later."},{"Start":"00:33.855 ","End":"00:36.955","Text":"Then we have dS,"},{"Start":"00:36.955 ","End":"00:40.220","Text":"which is for surface area."},{"Start":"00:40.220 ","End":"00:42.155","Text":"It\u0027s also sometimes written as dA."},{"Start":"00:42.155 ","End":"00:44.850","Text":"For now, we\u0027re going to use dS."},{"Start":"00:44.920 ","End":"00:50.915","Text":"Lastly, we have dV, for volume."},{"Start":"00:50.915 ","End":"00:52.595","Text":"Remember, l is for length,"},{"Start":"00:52.595 ","End":"00:54.830","Text":"S is for surface area of V for volume,"},{"Start":"00:54.830 ","End":"00:56.995","Text":"and it should all make a lot of sense."},{"Start":"00:56.995 ","End":"01:00.630","Text":"A quick note is that, as we measure our dl,"},{"Start":"01:00.630 ","End":"01:04.025","Text":"our dS, and our dV as we\u0027ll annotate them in this course."},{"Start":"01:04.025 ","End":"01:07.145","Text":"Is we\u0027re measuring, we call little length,"},{"Start":"01:07.145 ","End":"01:09.110","Text":"little surface area in little volume,"},{"Start":"01:09.110 ","End":"01:14.945","Text":"meaning it\u0027s an infinitesimal change in one of those elements, that we\u0027re measuring."},{"Start":"01:14.945 ","End":"01:18.885","Text":"We can assemble these with integrals to create a whole shape."},{"Start":"01:18.885 ","End":"01:21.460","Text":"If you want to give these measurements any real meaning,"},{"Start":"01:21.460 ","End":"01:25.545","Text":"we have to be measuring certain axes in the different systems."},{"Start":"01:25.545 ","End":"01:29.075","Text":"Obviously, based on the different coordinate system using,"},{"Start":"01:29.075 ","End":"01:30.610","Text":"the way you calculate that is little difference."},{"Start":"01:30.610 ","End":"01:32.120","Text":"That\u0027s really what we\u0027re exploring here."},{"Start":"01:32.120 ","End":"01:34.495","Text":"To start in Cartesian coordinates,"},{"Start":"01:34.495 ","End":"01:37.540","Text":"measuring each of these things is going to have its own formula, of course."},{"Start":"01:37.540 ","End":"01:39.850","Text":"Starting with length dl,"},{"Start":"01:39.850 ","End":"01:41.885","Text":"what we\u0027re measuring is really dx,"},{"Start":"01:41.885 ","End":"01:44.010","Text":"or dy, or dz."},{"Start":"01:44.010 ","End":"01:48.565","Text":"These lines are not for division."},{"Start":"01:48.565 ","End":"01:53.365","Text":"Just to tell you that you can do one or the other or the third option."},{"Start":"01:53.365 ","End":"01:59.200","Text":"Basically, what that means is dx is a line that is parallel to the x-axis."},{"Start":"01:59.200 ","End":"02:01.420","Text":"Infinitesimally small distance again,"},{"Start":"02:01.420 ","End":"02:05.575","Text":"dy is parallel to the y-axis,"},{"Start":"02:05.575 ","End":"02:11.050","Text":"and dV is a small distance parallel to the z-axis."},{"Start":"02:11.050 ","End":"02:15.530","Text":"We can take these distances and multiply them by each other to find our area."},{"Start":"02:15.530 ","End":"02:17.750","Text":"When we\u0027re finding our area,"},{"Start":"02:17.750 ","End":"02:19.895","Text":"we can do dx times dy,"},{"Start":"02:19.895 ","End":"02:23.370","Text":"or we can do dy times dz,"},{"Start":"02:23.370 ","End":"02:26.015","Text":"or we can do dz times dx."},{"Start":"02:26.015 ","End":"02:28.340","Text":"What that gives us is a multiplication"},{"Start":"02:28.340 ","End":"02:30.950","Text":"of one length by another length or width by length,"},{"Start":"02:30.950 ","End":"02:32.285","Text":"that gives us little square."},{"Start":"02:32.285 ","End":"02:34.505","Text":"For example, when we do dx by dy,"},{"Start":"02:34.505 ","End":"02:35.983","Text":"we can come up with square,"},{"Start":"02:35.983 ","End":"02:39.305","Text":"a little area that\u0027s parallel to the x-y plane."},{"Start":"02:39.305 ","End":"02:41.795","Text":"Similarly, when it\u0027s dy times dz,"},{"Start":"02:41.795 ","End":"02:46.955","Text":"we come up with a little area that is parallel to the dy dz plane."},{"Start":"02:46.955 ","End":"02:51.350","Text":"For dz dx you get the same thing on the x-z plane."},{"Start":"02:51.350 ","End":"02:53.525","Text":"Now if we\u0027re talking about dV,"},{"Start":"02:53.525 ","End":"02:54.650","Text":"we\u0027re talking in three dimensions."},{"Start":"02:54.650 ","End":"02:56.915","Text":"We do the same trick just with another dimension."},{"Start":"02:56.915 ","End":"03:00.440","Text":"We multiply all three of our quantities by each other,"},{"Start":"03:00.440 ","End":"03:03.095","Text":"dx by dy, dy by dz."},{"Start":"03:03.095 ","End":"03:04.940","Text":"The result is a little cube or"},{"Start":"03:04.940 ","End":"03:08.885","Text":"a little three dimensional area that shares all three elements."},{"Start":"03:08.885 ","End":"03:10.745","Text":"You can do with these cubes."},{"Start":"03:10.745 ","End":"03:13.452","Text":"Each one represents a little bit of volume, is you can,"},{"Start":"03:13.452 ","End":"03:15.035","Text":"through integrals,"},{"Start":"03:15.035 ","End":"03:19.325","Text":"have a number of them laid out and it\u0027ll give you the full volume of some shape."},{"Start":"03:19.325 ","End":"03:23.150","Text":"Now if we want to look at the cylindrical coordinate system,"},{"Start":"03:23.150 ","End":"03:25.580","Text":"we have three options when we\u0027re looking at lengths again,"},{"Start":"03:25.580 ","End":"03:27.290","Text":"we\u0027re going to start with length to begin with."},{"Start":"03:27.290 ","End":"03:28.910","Text":"You have rd Theta,"},{"Start":"03:28.910 ","End":"03:31.645","Text":"you have dr and you have dz."},{"Start":"03:31.645 ","End":"03:34.845","Text":"We can start with the easiest one dz,"},{"Start":"03:34.845 ","End":"03:37.775","Text":"is the same as the dz from our Cartesian coordinates,"},{"Start":"03:37.775 ","End":"03:39.080","Text":"just as we talked about in the last lecture."},{"Start":"03:39.080 ","End":"03:41.110","Text":"It\u0027s the same axis, it\u0027s the same dimension."},{"Start":"03:41.110 ","End":"03:42.485","Text":"You can measure it the same way."},{"Start":"03:42.485 ","End":"03:45.245","Text":"It\u0027s a line that\u0027s parallel to the z-axis."},{"Start":"03:45.245 ","End":"03:51.255","Text":"Dr is going to be similar to our Cartesian style d,"},{"Start":"03:51.255 ","End":"03:55.925","Text":"it\u0027s a line that\u0027s parallel to the radial axis to the r-axis."},{"Start":"03:55.925 ","End":"03:59.090","Text":"The new thing is rd Theta."},{"Start":"03:59.090 ","End":"04:03.095","Text":"Now rd Theta is really an arc and we\u0027re measuring a very short arc."},{"Start":"04:03.095 ","End":"04:07.370","Text":"You have to remember that when you want to find the arc of a circle with a"},{"Start":"04:07.370 ","End":"04:12.305","Text":"radius r. If you want to measure the length of that arc,"},{"Start":"04:12.305 ","End":"04:14.755","Text":"the measurement is r Theta."},{"Start":"04:14.755 ","End":"04:17.150","Text":"Theta being the angle that you\u0027re measuring."},{"Start":"04:17.150 ","End":"04:24.965","Text":"How we\u0027re going to apply that in the cylindrical coordinates,"},{"Start":"04:24.965 ","End":"04:30.005","Text":"is basically, we\u0027re measuring a very short angle Theta."},{"Start":"04:30.005 ","End":"04:34.580","Text":"It\u0027s at the point r which is the end of the radius."},{"Start":"04:34.580 ","End":"04:38.185","Text":"What we have is rd Theta."},{"Start":"04:38.185 ","End":"04:41.520","Text":"This is our first curve dimension,"},{"Start":"04:41.520 ","End":"04:44.350","Text":"and it\u0027s important to know because we\u0027re going to deal more"},{"Start":"04:44.350 ","End":"04:48.125","Text":"with the spherical coordinates."},{"Start":"04:48.125 ","End":"04:52.130","Text":"Basically, what it\u0027s for is if you are dealing with some ring,"},{"Start":"04:52.130 ","End":"04:56.755","Text":"some curved structure, and you want to break it down using integrals into small bits."},{"Start":"04:56.755 ","End":"05:00.520","Text":"It\u0027s easiest to do that to measure length in curved segments as opposed to"},{"Start":"05:00.520 ","End":"05:04.613","Text":"straight segments because it\u0027s then easier to assemble it later into the full shape."},{"Start":"05:04.613 ","End":"05:06.520","Text":"So you can measure what\u0027s going on."},{"Start":"05:06.520 ","End":"05:08.690","Text":"You can describe things more easily."},{"Start":"05:08.690 ","End":"05:10.625","Text":"Moving on to dS,"},{"Start":"05:10.625 ","End":"05:12.324","Text":"in the cylindrical coordinates,"},{"Start":"05:12.324 ","End":"05:14.500","Text":"you have the same three options that had before where you\u0027re"},{"Start":"05:14.500 ","End":"05:17.380","Text":"multiplying different lengths by each other."},{"Start":"05:17.380 ","End":"05:21.160","Text":"The first one we can start with is rd Theta dr. What do we do?"},{"Start":"05:21.160 ","End":"05:24.250","Text":"We take the arc rd Theta that we just talked about,"},{"Start":"05:24.250 ","End":"05:27.650","Text":"and we multiply it by r giving it a little bit of width in"},{"Start":"05:27.650 ","End":"05:31.625","Text":"the direction of the radial axis."},{"Start":"05:31.625 ","End":"05:34.190","Text":"The easiest way to describe that is you take a slice"},{"Start":"05:34.190 ","End":"05:37.745","Text":"here in an arc shape that expands as it goes."},{"Start":"05:37.745 ","End":"05:41.260","Text":"Another way to describe that is imagining of a slice of pizza,"},{"Start":"05:41.260 ","End":"05:44.240","Text":"and you want to describe just the crust of it,"},{"Start":"05:44.240 ","End":"05:47.465","Text":"or maybe a little slice from the middle of the pizza that you cut out like that."},{"Start":"05:47.465 ","End":"05:48.965","Text":"That\u0027s when we\u0027re going to use"},{"Start":"05:48.965 ","End":"05:54.245","Text":"our rd Theta dr. What you end up with is a little segment of a disk,"},{"Start":"05:54.245 ","End":"05:56.030","Text":"a little segment of a circle. It\u0027s going to be flat."},{"Start":"05:56.030 ","End":"06:02.339","Text":"It\u0027s going to be measured in area and it has the Theta element giving it the arc."},{"Start":"06:02.339 ","End":"06:04.955","Text":"It\u0027s measured with a width of dr."},{"Start":"06:04.955 ","End":"06:11.980","Text":"Instead of measuring just the perimeter of a circle as dl elements do, our rd Theta."},{"Start":"06:11.980 ","End":"06:14.210","Text":"It\u0027s actually measuring areas so you can use it"},{"Start":"06:14.210 ","End":"06:18.860","Text":"with integrals create a full circle and then describe that."},{"Start":"06:18.860 ","End":"06:23.610","Text":"When you\u0027re using it generally you\u0027re talking about rd Theta dr,"},{"Start":"06:23.610 ","End":"06:26.380","Text":"and Theta is going to be equal to 2Pi."},{"Start":"06:26.380 ","End":"06:28.140","Text":"That means that you\u0027re going to be measuring a full circle,"},{"Start":"06:28.140 ","End":"06:32.120","Text":"and you\u0027re going be putting together segments of it to describe the whole thing."},{"Start":"06:32.120 ","End":"06:34.280","Text":"You can, as you see here,"},{"Start":"06:34.280 ","End":"06:36.290","Text":"fill in the entire circle,"},{"Start":"06:36.290 ","End":"06:42.860","Text":"that would be rd2Pi dr. You\u0027re going to create a whole circle here."},{"Start":"06:42.860 ","End":"06:46.880","Text":"What you have is a small disk really, it\u0027s not a full circle."},{"Start":"06:46.880 ","End":"06:54.290","Text":"It\u0027s a disk, that\u0027s going to have the width of dr. Once you have one of them,"},{"Start":"06:54.290 ","End":"06:56.405","Text":"you can layer different bits on top of it,"},{"Start":"06:56.405 ","End":"07:00.290","Text":"one on top of the other to describe a full circle with no hole in the middle"},{"Start":"07:00.290 ","End":"07:04.880","Text":"or maybe something like a bagel that has a small hole in the middle."},{"Start":"07:04.880 ","End":"07:07.100","Text":"You can find your area that way."},{"Start":"07:07.100 ","End":"07:09.080","Text":"Also, it doesn\u0027t have to be a full circle."},{"Start":"07:09.080 ","End":"07:11.935","Text":"You could be describing something like our slice of pizza here."},{"Start":"07:11.935 ","End":"07:15.320","Text":"What you do is just instead of measuring Theta at 2Pi,"},{"Start":"07:15.320 ","End":"07:18.890","Text":"you find whatever the angle is at that relative to 2Pi."},{"Start":"07:18.890 ","End":"07:21.425","Text":"Maybe for our slice of pizza would be something like,"},{"Start":"07:21.425 ","End":"07:24.500","Text":"I don\u0027t know, 1/3 of Pi or 1/2 of Pi."},{"Start":"07:24.500 ","End":"07:27.850","Text":"You can measure that out to find the area."},{"Start":"07:27.850 ","End":"07:30.260","Text":"Now let\u0027s move on to our next element,"},{"Start":"07:30.260 ","End":"07:33.305","Text":"we\u0027ll do rd Theta dz."},{"Start":"07:33.305 ","End":"07:36.785","Text":"What we\u0027re doing is we\u0027re taking that same arc from before the rd Theta."},{"Start":"07:36.785 ","End":"07:38.390","Text":"Instead of expanding it with width,"},{"Start":"07:38.390 ","End":"07:39.680","Text":"we\u0027re giving it a little bit of height."},{"Start":"07:39.680 ","End":"07:43.984","Text":"You end up with as opposed to a square that we have in our Cartesian coordinates,"},{"Start":"07:43.984 ","End":"07:48.010","Text":"in cylindrical coordinates, you get a little slice of a cylinder basically."},{"Start":"07:48.010 ","End":"07:52.370","Text":"Imagine you\u0027re measuring the surface area of a paper towel roll or something like that."},{"Start":"07:52.370 ","End":"07:55.325","Text":"You want to find a little bit of the area there."},{"Start":"07:55.325 ","End":"07:59.250","Text":"That\u0027s what you\u0027re creating when you do rd Theta dz."},{"Start":"07:59.740 ","End":"08:02.060","Text":"When you use that,"},{"Start":"08:02.060 ","End":"08:06.230","Text":"what you\u0027re doing is instead of giving width to an arc, you\u0027re giving height to it."},{"Start":"08:06.230 ","End":"08:07.730","Text":"You can measure, as you see here,"},{"Start":"08:07.730 ","End":"08:12.380","Text":"a little segment around the cylindrical surface area."},{"Start":"08:12.380 ","End":"08:15.380","Text":"When you do integrals based on z,"},{"Start":"08:15.380 ","End":"08:19.570","Text":"you can add the entire height and it\u0027ll give you the whole surface area of that cylinder."},{"Start":"08:19.570 ","End":"08:23.570","Text":"Again, you can use it for paper towel roll if you\u0027re measuring,"},{"Start":"08:23.570 ","End":"08:25.085","Text":"I don\u0027t know, a telescope or something like that."},{"Start":"08:25.085 ","End":"08:27.380","Text":"It\u0027ll help you describe the area."},{"Start":"08:27.380 ","End":"08:29.150","Text":"Lastly, we have dr, dz."},{"Start":"08:29.150 ","End":"08:33.755","Text":"It\u0027s actually rather similar to the Cartesian style areas,"},{"Start":"08:33.755 ","End":"08:35.585","Text":"because you have dr,"},{"Start":"08:35.585 ","End":"08:38.240","Text":"which is a straight line parallel to r and dz,"},{"Start":"08:38.240 ","End":"08:40.295","Text":"which is straight line parallel to z."},{"Start":"08:40.295 ","End":"08:42.890","Text":"You\u0027re going to get a square. You\u0027re not going to get a curve in this,"},{"Start":"08:42.890 ","End":"08:45.260","Text":"it\u0027s going to be a squared segment but of course,"},{"Start":"08:45.260 ","End":"08:47.215","Text":"it\u0027s on the radial axis."},{"Start":"08:47.215 ","End":"08:51.050","Text":"So it\u0027s going to slice in the middle of your cylinder,"},{"Start":"08:51.050 ","End":"08:52.760","Text":"in the middle of your circle,"},{"Start":"08:52.760 ","End":"08:54.890","Text":"in the middle of whatever it is you\u0027re measuring,"},{"Start":"08:54.890 ","End":"08:58.520","Text":"with a little bit of height and a little bit of width without the curve in it."},{"Start":"08:58.520 ","End":"09:00.350","Text":"Now it\u0027s not particularly useful for"},{"Start":"09:00.350 ","End":"09:03.050","Text":"measuring things on a cylinder and more of the inside,"},{"Start":"09:03.050 ","End":"09:05.780","Text":"but it\u0027s still important to know what it is."},{"Start":"09:05.780 ","End":"09:08.660","Text":"Where it\u0027s most useful really is when we\u0027re talking about volume."},{"Start":"09:08.660 ","End":"09:10.130","Text":"If we skip to volume for a second,"},{"Start":"09:10.130 ","End":"09:13.115","Text":"we need our dr dz to calculate this because it is"},{"Start":"09:13.115 ","End":"09:16.760","Text":"the dr element times the dz element times the d Theta element."},{"Start":"09:16.760 ","End":"09:20.300","Text":"What we get as our formula is rd Theta dV dz."},{"Start":"09:20.300 ","End":"09:24.980","Text":"What we\u0027re doing is imagine you take the rd Theta dz element,"},{"Start":"09:24.980 ","End":"09:28.145","Text":"and give it a little bit of width from your dr element."},{"Start":"09:28.145 ","End":"09:31.130","Text":"What you get is a little chunk of a cylinder."},{"Start":"09:31.130 ","End":"09:33.280","Text":"Imagine if you had a deep dish pizza for example,"},{"Start":"09:33.280 ","End":"09:35.390","Text":"and someone wants to take a little square or"},{"Start":"09:35.390 ","End":"09:37.760","Text":"really cylindrical bit out of the middle of it."},{"Start":"09:37.760 ","End":"09:42.190","Text":"The beauty of this is you can do integration based on your Theta,"},{"Start":"09:42.190 ","End":"09:45.920","Text":"based on r and based on z and through any of these elements,"},{"Start":"09:45.920 ","End":"09:47.810","Text":"you can build a full cylinder."},{"Start":"09:47.810 ","End":"09:52.650","Text":"You can build a full cylindrical shape based on what your parameters are."},{"Start":"09:52.650 ","End":"09:54.530","Text":"You end up with volume,"},{"Start":"09:54.530 ","End":"09:56.570","Text":"of course, not just area, but the full volume."},{"Start":"09:56.570 ","End":"10:00.260","Text":"It\u0027s filled in, it\u0027s solid and it\u0027s useful."},{"Start":"10:00.260 ","End":"10:03.280","Text":"One last important thing with cylindrical coordinates is you\u0027re"},{"Start":"10:03.280 ","End":"10:05.810","Text":"going to want to remember what element it is that you\u0027re measuring,"},{"Start":"10:05.810 ","End":"10:07.820","Text":"what objects, which bit to use."},{"Start":"10:07.820 ","End":"10:14.980","Text":"For example, a CD-ROM disc you\u0027d want to use rd Theta dr. That\u0027s cylindrical coordinates."},{"Start":"10:14.980 ","End":"10:17.290","Text":"If we want to move on to spherical coordinates,"},{"Start":"10:17.290 ","End":"10:21.565","Text":"we can start with the length again dr,"},{"Start":"10:21.565 ","End":"10:24.535","Text":"is very similar to the dr from cylindrical coordinates."},{"Start":"10:24.535 ","End":"10:25.585","Text":"It\u0027s a straight line."},{"Start":"10:25.585 ","End":"10:30.460","Text":"Of course, this time it\u0027s coming from the origin and not from the z-axis."},{"Start":"10:30.460 ","End":"10:31.840","Text":"Again, it\u0027s very similar,"},{"Start":"10:31.840 ","End":"10:34.675","Text":"it\u0027s a straight line and there it is."},{"Start":"10:34.675 ","End":"10:37.405","Text":"Next, we can talk about rd Phi."},{"Start":"10:37.405 ","End":"10:38.740","Text":"It\u0027s going to be an arc again,"},{"Start":"10:38.740 ","End":"10:39.970","Text":"but this time it\u0027s going to be"},{"Start":"10:39.970 ","End":"10:43.240","Text":"a vertical style arc because if you look at how the fee angle changes."},{"Start":"10:43.240 ","End":"10:45.190","Text":"It\u0027s vertical, it\u0027s not horizontal."},{"Start":"10:45.190 ","End":"10:48.910","Text":"You end up with an arc that looks something like this,"},{"Start":"10:48.910 ","End":"10:53.020","Text":"where it\u0027s arching down from the z towards the x,"},{"Start":"10:53.020 ","End":"10:55.645","Text":"y plane, and it\u0027s a vertical change."},{"Start":"10:55.645 ","End":"10:59.065","Text":"Lastly, there\u0027s rd Phi."},{"Start":"10:59.065 ","End":"11:03.130","Text":"Lastly, we have r sine Phi d Theta."},{"Start":"11:03.130 ","End":"11:04.630","Text":"Why did we write it like that?"},{"Start":"11:04.630 ","End":"11:07.870","Text":"That\u0027s, again, the annotation we have for our x, y,"},{"Start":"11:07.870 ","End":"11:12.220","Text":"which is really the same as the measurement of r in the cylindrical coordinates."},{"Start":"11:12.220 ","End":"11:15.275","Text":"Here we write it as rxy if you remember from the last lecture."},{"Start":"11:15.275 ","End":"11:19.645","Text":"Rxy or r sine Phi is going to be"},{"Start":"11:19.645 ","End":"11:22.480","Text":"multiplied by Theta and you\u0027re going to get the same Theta curve"},{"Start":"11:22.480 ","End":"11:25.510","Text":"as you got in the cylindrical coordinates."},{"Start":"11:25.510 ","End":"11:29.995","Text":"In spherical coordinates, we just write it as r sine Phi d Theta,"},{"Start":"11:29.995 ","End":"11:32.710","Text":"but it really represents the same curve."},{"Start":"11:32.710 ","End":"11:35.530","Text":"It\u0027s important to remember here we have a horizontal curve,"},{"Start":"11:35.530 ","End":"11:36.910","Text":"and we have a vertical curve."},{"Start":"11:36.910 ","End":"11:39.624","Text":"This is actually what\u0027s really interesting about spherical coordinates."},{"Start":"11:39.624 ","End":"11:42.250","Text":"In reality, we don\u0027t look much at the dl elements."},{"Start":"11:42.250 ","End":"11:45.820","Text":"We don\u0027t look at length here because we can already describe them in the other 2 systems."},{"Start":"11:45.820 ","End":"11:51.205","Text":"What\u0027s more interesting is surface area and volume ds and dv."},{"Start":"11:51.205 ","End":"11:52.870","Text":"Even when we look at ds,"},{"Start":"11:52.870 ","End":"11:54.145","Text":"by the way, if you see here,"},{"Start":"11:54.145 ","End":"11:56.080","Text":"I\u0027ve only written out one of the formulas"},{"Start":"11:56.080 ","End":"11:58.405","Text":"because this is really the only one that really interests me."},{"Start":"11:58.405 ","End":"12:02.965","Text":"The other ones again, can be described through cylindrical or Cartesian coordinates."},{"Start":"12:02.965 ","End":"12:06.295","Text":"Forgetting for a second about the other two formulas."},{"Start":"12:06.295 ","End":"12:10.660","Text":"What really interests us here is when you want to find the surface area using"},{"Start":"12:10.660 ","End":"12:18.220","Text":"your 2 arcs elements when you try to use your rd Phi and your r sine Phi d Theta."},{"Start":"12:18.220 ","End":"12:21.070","Text":"What you do is again, we\u0027re multiplying the 2 elements by each other."},{"Start":"12:21.070 ","End":"12:26.695","Text":"You end up with r^2 sine Phi d Theta d Phi."},{"Start":"12:26.695 ","End":"12:28.210","Text":"When you put those together,"},{"Start":"12:28.210 ","End":"12:31.720","Text":"you get, a slice of area that\u0027s curved on both sides."},{"Start":"12:31.720 ","End":"12:35.530","Text":"It\u0027s like a bit of surface area of a sphere, of a globe."},{"Start":"12:35.530 ","End":"12:37.420","Text":"Think about it as maybe the peel of an orange,"},{"Start":"12:37.420 ","End":"12:39.895","Text":"if that helps you."},{"Start":"12:39.895 ","End":"12:43.360","Text":"What you\u0027re doing is taking a small segment of that surface area,"},{"Start":"12:43.360 ","End":"12:45.625","Text":"and you can, of course,"},{"Start":"12:45.625 ","End":"12:48.085","Text":"integrate based on your Theta element."},{"Start":"12:48.085 ","End":"12:52.915","Text":"What you get is a little ring around the surface area."},{"Start":"12:52.915 ","End":"12:54.130","Text":"If you think about it again as an orange,"},{"Start":"12:54.130 ","End":"12:57.805","Text":"it\u0027s like taking off 1 slice from the peel,"},{"Start":"12:57.805 ","End":"13:01.540","Text":"or it could be like the line along a baseball maybe."},{"Start":"13:01.540 ","End":"13:05.710","Text":"When you integrate based on the Phi element,"},{"Start":"13:05.710 ","End":"13:09.640","Text":"then you get the whole surface area of a sphere."},{"Start":"13:09.640 ","End":"13:11.515","Text":"Before we move on to volume,"},{"Start":"13:11.515 ","End":"13:13.120","Text":"there\u0027s one last thing I wanted to look at."},{"Start":"13:13.120 ","End":"13:18.145","Text":"It\u0027s d Omega. What Omega stands for is the sine phi d Theta d Phi."},{"Start":"13:18.145 ","End":"13:24.640","Text":"Basically, it\u0027s the entire angle of the surface area element here without the length."},{"Start":"13:24.640 ","End":"13:25.930","Text":"If we\u0027re talking about the angle,"},{"Start":"13:25.930 ","End":"13:28.660","Text":"you could describe it as an area angle,"},{"Start":"13:28.660 ","End":"13:32.140","Text":"not as an angle of a line, rather of an area."},{"Start":"13:32.140 ","End":"13:33.655","Text":"We\u0027ll talk about it more later,"},{"Start":"13:33.655 ","End":"13:36.295","Text":"but it\u0027s important to know what that is."},{"Start":"13:36.295 ","End":"13:38.545","Text":"Lastly, there\u0027s volume."},{"Start":"13:38.545 ","End":"13:40.300","Text":"What we\u0027re doing with volume is again,"},{"Start":"13:40.300 ","End":"13:42.940","Text":"multiplying our three elements one by the other."},{"Start":"13:42.940 ","End":"13:49.255","Text":"You end up with r^2 sine Phi dr d Theta d Phi."},{"Start":"13:49.255 ","End":"13:51.100","Text":"If you can imagine it this way,"},{"Start":"13:51.100 ","End":"13:54.010","Text":"we\u0027re taking the element we just talked about our surface area,"},{"Start":"13:54.010 ","End":"13:55.705","Text":"or the orange peel,"},{"Start":"13:55.705 ","End":"13:57.205","Text":"the surface area of the sphere,"},{"Start":"13:57.205 ","End":"14:00.250","Text":"and we\u0027re giving it some volume."},{"Start":"14:00.250 ","End":"14:06.025","Text":"If you see here, we have the Phi element and we have the sine Phi d Theta element."},{"Start":"14:06.025 ","End":"14:08.440","Text":"Now we\u0027re giving it a little bit of an r element to"},{"Start":"14:08.440 ","End":"14:10.915","Text":"give it some depth towards the center of our sphere,"},{"Start":"14:10.915 ","End":"14:12.640","Text":"to give it a little bit of width."},{"Start":"14:12.640 ","End":"14:14.560","Text":"What you can do with this is,"},{"Start":"14:14.560 ","End":"14:15.658","Text":"now that this has volume,"},{"Start":"14:15.658 ","End":"14:20.800","Text":"you can imagine it as more like a real orange peel,"},{"Start":"14:20.800 ","End":"14:22.705","Text":"in the sense it has some depth to it."},{"Start":"14:22.705 ","End":"14:29.845","Text":"You can take layer upon layer of this spherical shape,"},{"Start":"14:29.845 ","End":"14:32.445","Text":"and with it, you can create a full circle."},{"Start":"14:32.445 ","End":"14:36.855","Text":"As you can see here, imagine you\u0027re taking each layer upon itself like an onion,"},{"Start":"14:36.855 ","End":"14:39.825","Text":"for example, and building with it a sphere."},{"Start":"14:39.825 ","End":"14:43.140","Text":"Now, of course, this isn\u0027t just good for full spheres."},{"Start":"14:43.140 ","End":"14:45.060","Text":"You can also use it with different angles and"},{"Start":"14:45.060 ","End":"14:48.090","Text":"different measurements to measure a half sphere or a segment of a sphere."},{"Start":"14:48.090 ","End":"14:49.290","Text":"The same is true of course,"},{"Start":"14:49.290 ","End":"14:53.035","Text":"with the surface area doesn\u0027t have to be measuring the entire surface area of the sphere."},{"Start":"14:53.035 ","End":"14:54.730","Text":"You could be measuring part of the surface area,"},{"Start":"14:54.730 ","End":"14:57.070","Text":"surface area of an object that\u0027s part of a sphere."},{"Start":"14:57.070 ","End":"14:58.270","Text":"Of course, the same is true."},{"Start":"14:58.270 ","End":"15:02.050","Text":"Cylindrical objects doesn\u0027t have to be a full cylinder."},{"Start":"15:02.050 ","End":"15:04.675","Text":"It could be part of one in any given way,"},{"Start":"15:04.675 ","End":"15:09.700","Text":"or part of the disk surface area or part of the cylindrical surface area."},{"Start":"15:09.700 ","End":"15:13.150","Text":"Of course, there\u0027s other things we can measure in other coordinate systems,"},{"Start":"15:13.150 ","End":"15:14.530","Text":"but for now, we\u0027re going to leave it."},{"Start":"15:14.530 ","End":"15:19.225","Text":"One thing to mention now is a little bit about notation."},{"Start":"15:19.225 ","End":"15:21.610","Text":"There are two important things you want to talk about."},{"Start":"15:21.610 ","End":"15:25.060","Text":"One is this the rd Theta dr, and the other is,"},{"Start":"15:25.060 ","End":"15:29.965","Text":"what we just talked about, the volume formula for the spherical."},{"Start":"15:29.965 ","End":"15:34.705","Text":"Oftentimes people will write the rd Theta dr element out"},{"Start":"15:34.705 ","End":"15:40.330","Text":"already as 2 pi rdr. Why are they doing that?"},{"Start":"15:40.330 ","End":"15:45.460","Text":"Basically, they\u0027re doing a shortcut for what\u0027s going on here."},{"Start":"15:45.460 ","End":"15:47.320","Text":"Now, even with just one integral,"},{"Start":"15:47.320 ","End":"15:48.880","Text":"it\u0027s still a ds."},{"Start":"15:48.880 ","End":"15:53.020","Text":"It\u0027s still an integral of area because we have 2 units that are measured,"},{"Start":"15:53.020 ","End":"15:54.310","Text":"r is measured in length,"},{"Start":"15:54.310 ","End":"15:55.420","Text":"dr is measured in length."},{"Start":"15:55.420 ","End":"15:57.430","Text":"If you multiply them out in terms of the units,"},{"Start":"15:57.430 ","End":"15:59.740","Text":"we come out with an area measurement."},{"Start":"15:59.740 ","End":"16:02.935","Text":"Only do this in a case where I have an integral."},{"Start":"16:02.935 ","End":"16:07.180","Text":"Meaning, what I\u0027ll write out here is you have an integral that\u0027s based on"},{"Start":"16:07.180 ","End":"16:11.050","Text":"a function that is exclusively on r. On top of that,"},{"Start":"16:11.050 ","End":"16:16.675","Text":"we can do a second integral of the area that is on the rest of what\u0027s going on above,"},{"Start":"16:16.675 ","End":"16:20.590","Text":"rd Theta dr. Now the integral on Theta,"},{"Start":"16:20.590 ","End":"16:25.075","Text":"what we\u0027re going to do is do the entire angle from 0 until 2 Pi."},{"Start":"16:25.075 ","End":"16:26.995","Text":"With the second integral,"},{"Start":"16:26.995 ","End":"16:28.750","Text":"I know that it\u0027s not based on Theta."},{"Start":"16:28.750 ","End":"16:32.845","Text":"I know that the r is extraneous and so I can immediately go and do my integral in Theta."},{"Start":"16:32.845 ","End":"16:40.345","Text":"I have an integral of f(r) remaining and 2 Pi is my integral in theta times dr."},{"Start":"16:40.345 ","End":"16:44.605","Text":"In a situation where our function isn\u0027t determined by Theta rather by"},{"Start":"16:44.605 ","End":"16:49.465","Text":"r. We can immediately jump forward and determine our Theta to be equal to 2 Pi."},{"Start":"16:49.465 ","End":"16:53.035","Text":"Of course, this only works in a situation where we\u0027re measuring a full circle."},{"Start":"16:53.035 ","End":"16:54.625","Text":"We\u0027re measuring to 2 Pi."},{"Start":"16:54.625 ","End":"16:56.500","Text":"If we\u0027re only measuring half of a circle,"},{"Start":"16:56.500 ","End":"16:59.440","Text":"say 0 to 1 Pi or some other segment of a circle,"},{"Start":"16:59.440 ","End":"17:01.615","Text":"this doesn\u0027t work for us."},{"Start":"17:01.615 ","End":"17:05.935","Text":"Fortunately for us, in our course most of the time we are talking about full circles."},{"Start":"17:05.935 ","End":"17:09.595","Text":"A lot of times we can write r2 Pi dr,"},{"Start":"17:09.595 ","End":"17:12.925","Text":"from the get-go without having to go through the whole integral process."},{"Start":"17:12.925 ","End":"17:16.165","Text":"Because we know that if we\u0027re measuring a full circle, that\u0027s what we\u0027re going to do."},{"Start":"17:16.165 ","End":"17:19.360","Text":"Again, from the perspective of the shape that"},{"Start":"17:19.360 ","End":"17:22.630","Text":"you\u0027re getting of what you\u0027re really describing physically."},{"Start":"17:22.630 ","End":"17:26.470","Text":"This is going to give you some disk segment."},{"Start":"17:26.470 ","End":"17:29.500","Text":"If you recall before when we integrate based"},{"Start":"17:29.500 ","End":"17:32.140","Text":"on the Theta element, when you do the whole thing,"},{"Start":"17:32.140 ","End":"17:39.490","Text":"you end up with a disc segment with a depth or a width of dr. Of course,"},{"Start":"17:39.490 ","End":"17:41.920","Text":"once we have that, if we integrate based on r element,"},{"Start":"17:41.920 ","End":"17:43.645","Text":"we can get the full circle."},{"Start":"17:43.645 ","End":"17:45.880","Text":"Now, very similar thing is going to happen when we\u0027re"},{"Start":"17:45.880 ","End":"17:48.565","Text":"talking in our spherical coordinates to volume."},{"Start":"17:48.565 ","End":"17:52.270","Text":"In the same way that we did an integral on Theta before,"},{"Start":"17:52.270 ","End":"17:55.900","Text":"we\u0027re going to have an integral on Theta and one and Phi."},{"Start":"17:55.900 ","End":"17:58.840","Text":"The Theta integral, again,"},{"Start":"17:58.840 ","End":"18:02.620","Text":"we\u0027re going to write out sine Phi d Theta d Phi."},{"Start":"18:02.620 ","End":"18:07.180","Text":"The integral is going to be measuring the entire angle of Theta,"},{"Start":"18:07.180 ","End":"18:08.995","Text":"which is 0 to 2 Pi."},{"Start":"18:08.995 ","End":"18:11.740","Text":"The Phi integral measures from 0 to Pi,"},{"Start":"18:11.740 ","End":"18:14.635","Text":"the entire reach of the angle of Phi."},{"Start":"18:14.635 ","End":"18:18.310","Text":"When we do that, again,"},{"Start":"18:18.310 ","End":"18:20.680","Text":"this is in a situation where the function itself is based"},{"Start":"18:20.680 ","End":"18:23.575","Text":"on r and not on Phi and not on Theta, we get 4 Pi."},{"Start":"18:23.575 ","End":"18:25.480","Text":"We can write it out as follows."},{"Start":"18:25.480 ","End":"18:30.835","Text":"Integral on a function of r multiplied by dv are volume measurement."},{"Start":"18:30.835 ","End":"18:35.965","Text":"Then gets written as the integral of f,"},{"Start":"18:35.965 ","End":"18:40.915","Text":"function r times 4 Pi times r^2 times dr."},{"Start":"18:40.915 ","End":"18:49.480","Text":"This end result here of r4 r Pi r^2 dr is the same as our dv function from before."},{"Start":"18:49.480 ","End":"18:53.050","Text":"It\u0027s just once we\u0027ve done the integral on Pi and on Phi."},{"Start":"18:53.050 ","End":"18:54.640","Text":"Of course, only in a situation where we"},{"Start":"18:54.640 ","End":"18:56.710","Text":"truly are measuring the full angle of both of them,"},{"Start":"18:56.710 ","End":"19:01.855","Text":"we\u0027re measuring a full sphere and not a hemisphere or some segment of it."},{"Start":"19:01.855 ","End":"19:03.550","Text":"When we draw this out,"},{"Start":"19:03.550 ","End":"19:05.545","Text":"what we get is what we talked about before."},{"Start":"19:05.545 ","End":"19:07.810","Text":"The peel of an orange,"},{"Start":"19:07.810 ","End":"19:11.125","Text":"a layer of an onion, some spherical layer."},{"Start":"19:11.125 ","End":"19:14.560","Text":"That\u0027s once you\u0027ve integrated it based on Phi and Theta,"},{"Start":"19:14.560 ","End":"19:16.870","Text":"and then if we integrate on r,"},{"Start":"19:16.870 ","End":"19:22.180","Text":"we\u0027re going to get layer upon layer upon layer and fill in the entire sphere."},{"Start":"19:22.180 ","End":"19:25.735","Text":"One last thing when we\u0027re talking about these two shortcuts is that,"},{"Start":"19:25.735 ","End":"19:29.545","Text":"that is the same as dv and that is the same as ds,"},{"Start":"19:29.545 ","End":"19:32.920","Text":"even though there\u0027s only 1 integral."},{"Start":"19:32.920 ","End":"19:37.360","Text":"One last thing, I didn\u0027t really talk about how we end up with rd Theta,"},{"Start":"19:37.360 ","End":"19:41.500","Text":"for example, as a measurement of lengths or any of the other measurements of length."},{"Start":"19:41.500 ","End":"19:43.315","Text":"A long story short,"},{"Start":"19:43.315 ","End":"19:46.975","Text":"there is a term that you may or may not be familiar with called the Jacobian matrix,"},{"Start":"19:46.975 ","End":"19:48.780","Text":"and we\u0027ve already incorporated it here."},{"Start":"19:48.780 ","End":"19:51.780","Text":"That\u0027s what you get is the r before you get to d Theta or the"},{"Start":"19:51.780 ","End":"19:55.920","Text":"sine Phi before you get to d Theta in volume, for example."},{"Start":"19:55.920 ","End":"19:58.680","Text":"It\u0027s something that we\u0027ve already incorporated, it\u0027s accounted for here."},{"Start":"19:58.680 ","End":"20:00.630","Text":"Those of you who are familiar with it, you don\u0027t have to worry about it."},{"Start":"20:00.630 ","End":"20:01.800","Text":"Those who are unfamiliar with it,"},{"Start":"20:01.800 ","End":"20:05.605","Text":"you\u0027ll learn about it later but long story short,"},{"Start":"20:05.605 ","End":"20:09.430","Text":"it\u0027s accounted for and this is how we have our length measurements."},{"Start":"20:09.430 ","End":"20:12.650","Text":"Anyway, that\u0027s the end of the lecture. Thank you for listening."}],"ID":9187},{"Watched":false,"Name":"Exercise - Circular Area","Duration":"4m 43s","ChapterTopicVideoID":8914,"CourseChapterTopicPlaylistID":144734,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8914.jpeg","UploadDate":"2017-03-21T08:22:42.2730000","DurationForVideoObject":"PT4M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.425","Text":"This problem is asking for us to get the area of a disc."},{"Start":"00:04.425 ","End":"00:08.025","Text":"Really we\u0027re calculating the area of a circle in the end because it\u0027s a full disc."},{"Start":"00:08.025 ","End":"00:11.385","Text":"You can see that because our answer is Pi R^2."},{"Start":"00:11.385 ","End":"00:14.580","Text":"What we\u0027re going to do is we can do that using integrals of"},{"Start":"00:14.580 ","End":"00:17.205","Text":"an area element in cylindrical coordinates."},{"Start":"00:17.205 ","End":"00:19.530","Text":"The full surface area,"},{"Start":"00:19.530 ","End":"00:23.295","Text":"the full S that we\u0027re trying to find is the integral of dS."},{"Start":"00:23.295 ","End":"00:25.860","Text":"If you recall dS is basically when"},{"Start":"00:25.860 ","End":"00:28.650","Text":"you take 2 length elements and multiply them by each other."},{"Start":"00:28.650 ","End":"00:30.225","Text":"If we have our circle here,"},{"Start":"00:30.225 ","End":"00:32.160","Text":"which is our potential area,"},{"Start":"00:32.160 ","End":"00:36.140","Text":"we have the large radius R of the full circle and we have"},{"Start":"00:36.140 ","End":"00:41.528","Text":"a little area element here which is made up of dr and dTheta,"},{"Start":"00:41.528 ","End":"00:45.045","Text":"dr being the change in our r axis,"},{"Start":"00:45.045 ","End":"00:48.650","Text":"the width of it, and the dTheta being the curved portion,"},{"Start":"00:48.650 ","End":"00:51.020","Text":"the arc, what gives it its circular form."},{"Start":"00:51.020 ","End":"00:54.650","Text":"The way we take this area element and turn into the full area of the circle is"},{"Start":"00:54.650 ","End":"00:58.580","Text":"first we do an integral based on the Theta element."},{"Start":"00:58.580 ","End":"01:01.160","Text":"We take it all the way around and turn this little segment into"},{"Start":"01:01.160 ","End":"01:04.050","Text":"a full ring in the middle of our circle."},{"Start":"01:04.050 ","End":"01:09.350","Text":"Then what we do as we can do another integral on the radius to big R,"},{"Start":"01:09.350 ","End":"01:13.370","Text":"which basically gives us little layers of that circle and fills in the entire thing until"},{"Start":"01:13.370 ","End":"01:17.870","Text":"we have all the way out to the edge to R. What we have here,"},{"Start":"01:17.870 ","End":"01:20.390","Text":"mathwise is a double integral."},{"Start":"01:20.390 ","End":"01:21.530","Text":"We have to do 2 integrals here,"},{"Start":"01:21.530 ","End":"01:24.200","Text":"one on dr and one on dTheta."},{"Start":"01:24.200 ","End":"01:28.670","Text":"What that means for us is it\u0027s exactly what we described here,"},{"Start":"01:28.670 ","End":"01:30.750","Text":"is that you\u0027re first doing the integral on"},{"Start":"01:30.750 ","End":"01:33.290","Text":"the Theta element to give you the ring and then you\u0027re doing an"},{"Start":"01:33.290 ","End":"01:35.780","Text":"integral on the r element to expand that"},{"Start":"01:35.780 ","End":"01:39.505","Text":"ring outwards to get the entire radius of the circle."},{"Start":"01:39.505 ","End":"01:42.950","Text":"What we\u0027re going to do with that is, first of all,"},{"Start":"01:42.950 ","End":"01:45.275","Text":"you need to know that all of our limits here,"},{"Start":"01:45.275 ","End":"01:49.025","Text":"our constants, they\u0027re constant, they\u0027re not variables."},{"Start":"01:49.025 ","End":"01:51.950","Text":"Because of that the order of this doesn\u0027t really matter."},{"Start":"01:51.950 ","End":"01:54.320","Text":"Instead of writing it as it\u0027s written here,"},{"Start":"01:54.320 ","End":"01:55.700","Text":"we could have also done the opposite."},{"Start":"01:55.700 ","End":"01:58.475","Text":"We could have written integral 0-R,"},{"Start":"01:58.475 ","End":"02:01.940","Text":"integral 0-2Pi and then rearrange the order inside as well,"},{"Start":"02:01.940 ","End":"02:05.030","Text":"rd Theta dr. Now most of the time in"},{"Start":"02:05.030 ","End":"02:08.600","Text":"our course you\u0027re going to deal with limits that are constant."},{"Start":"02:08.600 ","End":"02:12.650","Text":"However, if you were dealing with a limit that was a variable,"},{"Start":"02:12.650 ","End":"02:14.990","Text":"the order does matter."},{"Start":"02:14.990 ","End":"02:22.385","Text":"For example, you could have the limit instead of 2Pi being little r over large R that is."},{"Start":"02:22.385 ","End":"02:26.965","Text":"Some variable that\u0027s determined by what\u0027s inside of the formula."},{"Start":"02:26.965 ","End":"02:30.040","Text":"In that case, you would have to do the little r integral"},{"Start":"02:30.040 ","End":"02:33.790","Text":"first and the constant integral second."},{"Start":"02:33.790 ","End":"02:36.850","Text":"When it comes time to actually calculate the integral,"},{"Start":"02:36.850 ","End":"02:38.800","Text":"imagine that you have some parentheses here,"},{"Start":"02:38.800 ","End":"02:42.400","Text":"that basically the 0-R integral is"},{"Start":"02:42.400 ","End":"02:47.385","Text":"inside parentheses with the r dr and that\u0027s your first function,"},{"Start":"02:47.385 ","End":"02:49.710","Text":"and that you\u0027re 0-2Pi,"},{"Start":"02:49.710 ","End":"02:53.055","Text":"integral goes to the last differential."},{"Start":"02:53.055 ","End":"02:55.410","Text":"Imagine you\u0027re having some parentheses here,"},{"Start":"02:55.410 ","End":"02:58.840","Text":"it\u0027s as though the inner function is happening first and the"},{"Start":"02:58.840 ","End":"03:02.470","Text":"integral that\u0027s outside is happening second with the outer terms."},{"Start":"03:02.470 ","End":"03:04.710","Text":"Inside and inside, outside and outside."},{"Start":"03:04.710 ","End":"03:06.130","Text":"Now, when you actually calculate this,"},{"Start":"03:06.130 ","End":"03:08.110","Text":"the best way to do it is to think of it as"},{"Start":"03:08.110 ","End":"03:13.045","Text":"2 different integrals and start with the first one, the inside one."},{"Start":"03:13.045 ","End":"03:16.955","Text":"If we take this integral and calculate it out,"},{"Start":"03:16.955 ","End":"03:22.295","Text":"what we end up with is r^2 over 2 and we have to insert the limits,"},{"Start":"03:22.295 ","End":"03:25.610","Text":"0-R. We\u0027re still left with 0-2Pi,"},{"Start":"03:25.610 ","End":"03:27.050","Text":"dTheta on the side."},{"Start":"03:27.050 ","End":"03:28.685","Text":"But again, we\u0027re not dealing with Theta yet."},{"Start":"03:28.685 ","End":"03:31.190","Text":"Only once we entirely solved the first integral,"},{"Start":"03:31.190 ","End":"03:32.965","Text":"we move on to the second one."},{"Start":"03:32.965 ","End":"03:36.590","Text":"When we take those limits and insert them, 0^2 over 2,"},{"Start":"03:36.590 ","End":"03:42.035","Text":"of course is 0 and r^2 over 2 is large R^2."},{"Start":"03:42.035 ","End":"03:44.495","Text":"We end up with the following here."},{"Start":"03:44.495 ","End":"03:47.355","Text":"We have an integral of 0-2Pi,"},{"Start":"03:47.355 ","End":"03:50.190","Text":"r^2 over 2, and dTheta."},{"Start":"03:50.190 ","End":"03:52.810","Text":"Now that we\u0027ve solved our inner integral,"},{"Start":"03:52.810 ","End":"03:54.640","Text":"we can work on our outer integral."},{"Start":"03:54.640 ","End":"03:58.120","Text":"We have our R^2 over 2, that\u0027s a constant."},{"Start":"03:58.120 ","End":"03:59.440","Text":"Now there\u0027s no more differential there,"},{"Start":"03:59.440 ","End":"04:00.819","Text":"so we can take it to the outside."},{"Start":"04:00.819 ","End":"04:05.335","Text":"You see it\u0027s R^2 over 2 multiplied by the remaining integral."},{"Start":"04:05.335 ","End":"04:07.870","Text":"If we do that integral, the d falls out,"},{"Start":"04:07.870 ","End":"04:11.600","Text":"and we end up with Theta with the limits of 0-2Pi."},{"Start":"04:11.600 ","End":"04:13.420","Text":"If you insert the values there,"},{"Start":"04:13.420 ","End":"04:17.110","Text":"you end up with 2Pi times R^ 2 over 2."},{"Start":"04:17.110 ","End":"04:19.735","Text":"If you simplify that, you end up with PiR^2."},{"Start":"04:19.735 ","End":"04:24.215","Text":"There you go, you\u0027ve got your solution and this is a great way to find areas."},{"Start":"04:24.215 ","End":"04:28.030","Text":"Now what you can think about is how you would do the same calculation for a segment"},{"Start":"04:28.030 ","End":"04:32.230","Text":"instead of a full circle with a given angle of Theta_0."},{"Start":"04:32.230 ","End":"04:36.645","Text":"It\u0027s the same operation except when you\u0027re doing integral instead of a limit to 2Pi,"},{"Start":"04:36.645 ","End":"04:39.510","Text":"you do 0 to Theta_0."},{"Start":"04:39.510 ","End":"04:42.070","Text":"That ends this problem."}],"ID":9188},{"Watched":false,"Name":"Exercise - Calculating Cylindrical Volume","Duration":"2m 34s","ChapterTopicVideoID":8915,"CourseChapterTopicPlaylistID":144734,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8915.jpeg","UploadDate":"2021-11-18T08:37:13.9270000","DurationForVideoObject":"PT2M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.075","Text":"In this problem, we want to do something similar to in our last one."},{"Start":"00:03.075 ","End":"00:06.285","Text":"This time we\u0027re finding volume in a cylinder,"},{"Start":"00:06.285 ","End":"00:09.500","Text":"and we want to do that is we\u0027re looking for V, as you can see,"},{"Start":"00:09.500 ","End":"00:12.690","Text":"and we know already that because it\u0027s cylindrical volume,"},{"Start":"00:12.690 ","End":"00:20.100","Text":"it\u0027s going to be Pi R^2 times height h. The way we do that is we take an integral of dV,"},{"Start":"00:20.100 ","End":"00:25.170","Text":"our infinitesimally small value for our volume segment within our cylinder."},{"Start":"00:25.170 ","End":"00:28.395","Text":"If you recall, if we draw our little cylinder out,"},{"Start":"00:28.395 ","End":"00:31.830","Text":"it looks like a small segment taken out that has some height."},{"Start":"00:31.830 ","End":"00:33.615","Text":"It has some arc or length to it."},{"Start":"00:33.615 ","End":"00:35.460","Text":"It has some depth or radius to it,"},{"Start":"00:35.460 ","End":"00:37.575","Text":"it has all 3 dimensions, it\u0027s a small segment."},{"Start":"00:37.575 ","End":"00:41.220","Text":"What we\u0027re going to do is, using all 3 of the functions we saw earlier,"},{"Start":"00:41.220 ","End":"00:44.075","Text":"we\u0027re going to combine them and do integrals on each of those"},{"Start":"00:44.075 ","End":"00:48.125","Text":"to expand out our segment to cover the entire shape."},{"Start":"00:48.125 ","End":"00:51.720","Text":"We\u0027re going to do dR, dTheta,"},{"Start":"00:51.720 ","End":"00:55.880","Text":"and dz on all of these integrals,"},{"Start":"00:55.880 ","End":"00:58.660","Text":"we\u0027re going to add dimension to it."},{"Start":"00:58.660 ","End":"01:04.440","Text":"Because all of our integrals are limited by constants,"},{"Start":"01:04.440 ","End":"01:06.750","Text":"0-r, 0-2Pi,"},{"Start":"01:06.750 ","End":"01:12.365","Text":"and 0-h, we can do our integrals just one after the other,"},{"Start":"01:12.365 ","End":"01:15.680","Text":"separately from each other on the same way that we did in our last problem."},{"Start":"01:15.680 ","End":"01:19.430","Text":"If you recall, we can imagine there\u0027s parentheses around each set of integrals."},{"Start":"01:19.430 ","End":"01:22.320","Text":"For example, the 0-r integral is"},{"Start":"01:22.320 ","End":"01:25.550","Text":"for the rdr and that is within its own set of parentheses,"},{"Start":"01:25.550 ","End":"01:31.895","Text":"meaning the inside integral in the inside term go with each other."},{"Start":"01:31.895 ","End":"01:36.440","Text":"The middle integral and the middle term go with each other,"},{"Start":"01:36.440 ","End":"01:39.500","Text":"and the outside integral and the outside term go with each other."},{"Start":"01:39.500 ","End":"01:42.650","Text":"If we start to calculate this, we\u0027ll start getting some results."},{"Start":"01:42.650 ","End":"01:50.250","Text":"We know from before that the integral of rdr from 0-r is r^2 over 2."},{"Start":"01:51.890 ","End":"01:55.130","Text":"With the values 0-r,"},{"Start":"01:55.130 ","End":"01:56.945","Text":"and if we plug in the limits,"},{"Start":"01:56.945 ","End":"01:59.020","Text":"we get r^2 over 2."},{"Start":"01:59.020 ","End":"02:03.080","Text":"Then we still have the integral of 0-2Pi of dTheta and dz."},{"Start":"02:03.080 ","End":"02:04.310","Text":"We\u0027re not going do with dz right now."},{"Start":"02:04.310 ","End":"02:07.985","Text":"We\u0027re just going to do the dTheta integral and we\u0027ll leave that dz for later."},{"Start":"02:07.985 ","End":"02:11.720","Text":"If we recall the integral of 0-2Pi of dTheta"},{"Start":"02:11.720 ","End":"02:16.070","Text":"is Theta with a limit of 0-2Pi which ends up being 2Pi."},{"Start":"02:16.070 ","End":"02:19.610","Text":"Now we have r^2 over 2 and 2Pi on the outside,"},{"Start":"02:19.610 ","End":"02:24.360","Text":"if you see here, and a z integral from 0-h still in the inside."},{"Start":"02:24.360 ","End":"02:27.890","Text":"So z from 0-h is h. As you see,"},{"Start":"02:27.890 ","End":"02:29.060","Text":"we end up with our final result,"},{"Start":"02:29.060 ","End":"02:32.765","Text":"Pi R^2 multiplied by h, the height."},{"Start":"02:32.765 ","End":"02:35.280","Text":"There you have it. You\u0027ve found your volume."}],"ID":9189}],"Thumbnail":null,"ID":144734},{"Name":"Density","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Density","Duration":"6m 4s","ChapterTopicVideoID":8916,"CourseChapterTopicPlaylistID":144735,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8916.jpeg","UploadDate":"2017-03-21T08:24:56.7370000","DurationForVideoObject":"PT6M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.420","Text":"In this lecture, I\u0027d like to define density."},{"Start":"00:03.420 ","End":"00:06.630","Text":"We can think of density in terms of"},{"Start":"00:06.630 ","End":"00:10.680","Text":"the units of mass or the quantity of mass in a given unit of volume."},{"Start":"00:10.680 ","End":"00:13.140","Text":"If we\u0027re talking about volume density,"},{"Start":"00:13.140 ","End":"00:16.035","Text":"that\u0027s symbolized by the Greek symbol Rho."},{"Start":"00:16.035 ","End":"00:22.695","Text":"The way we can calculate this is if we assume that in our cube on the side here,"},{"Start":"00:22.695 ","End":"00:26.805","Text":"we have 60 grams of mass,"},{"Start":"00:26.805 ","End":"00:30.135","Text":"m equals 60 grams there."},{"Start":"00:30.135 ","End":"00:32.265","Text":"One side of our cube,"},{"Start":"00:32.265 ","End":"00:34.275","Text":"its height is 1 centimeter,"},{"Start":"00:34.275 ","End":"00:36.540","Text":"the depth is 2 centimeters,"},{"Start":"00:36.540 ","End":"00:39.950","Text":"and the width is 3 centimeters, let\u0027s say."},{"Start":"00:39.950 ","End":"00:41.990","Text":"Then we can calculate the volume."},{"Start":"00:41.990 ","End":"00:44.855","Text":"The volume is 1 times 2 times 3,"},{"Start":"00:44.855 ","End":"00:47.665","Text":"is 6 centimeters to the third."},{"Start":"00:47.665 ","End":"00:50.100","Text":"Now with our m and our v, we can calculate Rho,"},{"Start":"00:50.100 ","End":"00:53.100","Text":"which is our volume density."},{"Start":"00:53.100 ","End":"00:57.675","Text":"So 60 grams over 6 centimeters to the"},{"Start":"00:57.675 ","End":"01:04.640","Text":"third is going to be equal to 10 grams per cubic centimeter to the third."},{"Start":"01:04.640 ","End":"01:07.235","Text":"Now if we\u0027re talking about what density really is,"},{"Start":"01:07.235 ","End":"01:11.480","Text":"you can think about that in terms of how much mass you have in a given part,"},{"Start":"01:11.480 ","End":"01:16.050","Text":"in a given unit of a cube or of any set of volume."},{"Start":"01:16.050 ","End":"01:20.480","Text":"If we color in that tiny segment of our larger cube,"},{"Start":"01:20.480 ","End":"01:25.165","Text":"we can say if that were maybe 1 centimeter squared,"},{"Start":"01:25.165 ","End":"01:27.175","Text":"we would have 60 grams of mass."},{"Start":"01:27.175 ","End":"01:29.115","Text":"That\u0027s one way to think about it,"},{"Start":"01:29.115 ","End":"01:32.300","Text":"what\u0027s the quantity or units of mass in a given unit of volume."},{"Start":"01:32.300 ","End":"01:35.590","Text":"Now if we\u0027re talking about area density, we\u0027re going to use Sigma."},{"Start":"01:35.590 ","End":"01:37.460","Text":"Some people use Rho for everything."},{"Start":"01:37.460 ","End":"01:40.295","Text":"I\u0027m going to make sure to use Sigma when we\u0027re talking about area density."},{"Start":"01:40.295 ","End":"01:42.380","Text":"We\u0027re going to use Lambda when we\u0027re talking about linear density,"},{"Start":"01:42.380 ","End":"01:45.025","Text":"we\u0027re going to use Rho only when we talk about volume density."},{"Start":"01:45.025 ","End":"01:47.360","Text":"Area density, it\u0027s the same idea."},{"Start":"01:47.360 ","End":"01:49.220","Text":"We want to take the mass,"},{"Start":"01:49.220 ","End":"01:55.325","Text":"quantity of mass in a unit of area as opposed to a unit of volume. How do we do that?"},{"Start":"01:55.325 ","End":"02:01.505","Text":"Let\u0027s assume that the mass of this plane is 2 kilograms"},{"Start":"02:01.505 ","End":"02:09.120","Text":"and the dimensions of it are the width is 1 meter and the length is 2 meters,"},{"Start":"02:09.120 ","End":"02:13.020","Text":"so our area is going to be 1 by 2,"},{"Start":"02:13.020 ","End":"02:14.660","Text":"which is 2 meters squared."},{"Start":"02:14.660 ","End":"02:16.910","Text":"Our volume, we already know to be 2 kilograms,"},{"Start":"02:16.910 ","End":"02:22.325","Text":"so it\u0027s 2 over 2 kilograms per meters squared that is."},{"Start":"02:22.325 ","End":"02:25.325","Text":"We end up with 1 kilogram per meter squared."},{"Start":"02:25.325 ","End":"02:28.130","Text":"The same way that we talked about mass per unit of volume above,"},{"Start":"02:28.130 ","End":"02:31.820","Text":"we can do the same per unit of area here cutting"},{"Start":"02:31.820 ","End":"02:35.960","Text":"out bits and saying that that bit has a certain amount of weight in it,"},{"Start":"02:35.960 ","End":"02:37.835","Text":"certain amount of mass in it that is."},{"Start":"02:37.835 ","End":"02:39.740","Text":"Now if we look at linear density,"},{"Start":"02:39.740 ","End":"02:42.620","Text":"this is the density of something that\u0027s a line, that\u0027s a length."},{"Start":"02:42.620 ","End":"02:44.090","Text":"It\u0027s the same concept again,"},{"Start":"02:44.090 ","End":"02:46.745","Text":"quantity of mass in a given line or unit of length."},{"Start":"02:46.745 ","End":"02:52.865","Text":"We take the mass over the length l. Let\u0027s say that we have this steak here"},{"Start":"02:52.865 ","End":"03:00.125","Text":"and it has a given unit there that we want to find what the mass is."},{"Start":"03:00.125 ","End":"03:03.650","Text":"If we say that the mass is overall in the entire steak"},{"Start":"03:03.650 ","End":"03:07.535","Text":"2 kilograms and the length of the steak is 1 meter,"},{"Start":"03:07.535 ","End":"03:13.625","Text":"then we know that the density Lambda is 2 kilograms per meter."},{"Start":"03:13.625 ","End":"03:17.135","Text":"We can also look at that in different units."},{"Start":"03:17.135 ","End":"03:22.145","Text":"Let\u0027s say that we want to find the mass of that 1-centimeter track there in the middle."},{"Start":"03:22.145 ","End":"03:25.070","Text":"We would do everything in grams to centimeters."},{"Start":"03:25.070 ","End":"03:26.420","Text":"Say it\u0027s 2,000 grams,"},{"Start":"03:26.420 ","End":"03:27.860","Text":"that is the same as 2 kilograms,"},{"Start":"03:27.860 ","End":"03:30.260","Text":"and 100 centimeters is the same as 1 meter."},{"Start":"03:30.260 ","End":"03:32.270","Text":"We can say the density is also 2,000"},{"Start":"03:32.270 ","End":"03:36.110","Text":"grams per 100 centimeters or 20 grams per centimeter."},{"Start":"03:36.110 ","End":"03:38.300","Text":"If we look at that little segment we cut out,"},{"Start":"03:38.300 ","End":"03:39.470","Text":"if that was 1 centimeter,"},{"Start":"03:39.470 ","End":"03:46.360","Text":"we could say that there\u0027s 20 grams of mass in that piece there."},{"Start":"03:46.490 ","End":"03:50.255","Text":"When are you going to use these formulas for density?"},{"Start":"03:50.255 ","End":"03:53.360","Text":"Ultimately, density is going to help you measure mass when you"},{"Start":"03:53.360 ","End":"03:56.360","Text":"have a given volume and you don\u0027t know the mass of it."},{"Start":"03:56.360 ","End":"04:01.175","Text":"Let\u0027s go back to the example above the volume density."},{"Start":"04:01.175 ","End":"04:02.990","Text":"If we know that this square,"},{"Start":"04:02.990 ","End":"04:07.280","Text":"let\u0027s say it\u0027s a block of cheese and we want to cut out a slice of it,"},{"Start":"04:07.280 ","End":"04:11.390","Text":"we can find out exactly what the mass is if we know the volume even if"},{"Start":"04:11.390 ","End":"04:15.760","Text":"we don\u0027t have the means to weigh the smaller cube that we cut out."},{"Start":"04:15.760 ","End":"04:18.410","Text":"We know that the mass of this is 60 grams."},{"Start":"04:18.410 ","End":"04:23.660","Text":"We know that the overall volume is 6 centimeters to the third, 6 cubic centimeters."},{"Start":"04:23.660 ","End":"04:25.595","Text":"If we cut off that little bit and say,"},{"Start":"04:25.595 ","End":"04:30.185","Text":"the volume of that cube that we\u0027re cutting out is 2 cubic centimeters,"},{"Start":"04:30.185 ","End":"04:32.825","Text":"then we can, with our given row,"},{"Start":"04:32.825 ","End":"04:34.730","Text":"find what the mass of that is."},{"Start":"04:34.730 ","End":"04:37.010","Text":"What we do is we take m,"},{"Start":"04:37.010 ","End":"04:40.920","Text":"which is the mass of our new object, equals Rho,"},{"Start":"04:40.920 ","End":"04:43.210","Text":"our density constant,"},{"Start":"04:43.210 ","End":"04:48.965","Text":"times v, which is in this case 2 cubic centimeters,"},{"Start":"04:48.965 ","End":"04:51.725","Text":"again v of our new object."},{"Start":"04:51.725 ","End":"04:53.915","Text":"We can go ahead and calculate that."},{"Start":"04:53.915 ","End":"04:58.340","Text":"We know that if it\u0027s 60 grams per 6 cubic centimeters,"},{"Start":"04:58.340 ","End":"05:01.849","Text":"we found out that it\u0027s 10 grams per cubic centimeter is our density,"},{"Start":"05:01.849 ","End":"05:07.650","Text":"so Rho equals 10 grams per cubic centimeter."},{"Start":"05:07.650 ","End":"05:15.980","Text":"If we multiply that 10 grams per cubic centimeter by 2 cubic centimeters,"},{"Start":"05:15.980 ","End":"05:19.690","Text":"the cubic centimeters drop out and we\u0027re left with only gram units."},{"Start":"05:19.690 ","End":"05:26.345","Text":"We know that our 2 cubic centimeter bit of cheese is 20 grams in mass."},{"Start":"05:26.345 ","End":"05:28.520","Text":"Now, of course, the same applies to area,"},{"Start":"05:28.520 ","End":"05:30.940","Text":"the same applies to linear density. We can do the same thing."},{"Start":"05:30.940 ","End":"05:32.360","Text":"If we cut out a small bit,"},{"Start":"05:32.360 ","End":"05:35.030","Text":"if we have a consistent density throughout the entire shape,"},{"Start":"05:35.030 ","End":"05:38.120","Text":"we can really find what we\u0027re looking for in terms of the mass."},{"Start":"05:38.120 ","End":"05:41.825","Text":"Now, again, that\u0027s if we have consistent density."},{"Start":"05:41.825 ","End":"05:44.225","Text":"If we do not have consistent density,"},{"Start":"05:44.225 ","End":"05:46.295","Text":"it\u0027s hard to find it."},{"Start":"05:46.295 ","End":"05:48.470","Text":"Usually, it will be given to you in the problem if you have"},{"Start":"05:48.470 ","End":"05:52.025","Text":"an inconsistent density throughout some shape, some object."},{"Start":"05:52.025 ","End":"05:56.900","Text":"For now, just remember these formulas for your consistent density with Rho,"},{"Start":"05:56.900 ","End":"06:00.900","Text":"Sigma, and Lambda and you should be fine. Thanks for listening."}],"ID":9190},{"Watched":false,"Name":"Exercise- Disc With a Hole","Duration":"1m 18s","ChapterTopicVideoID":8917,"CourseChapterTopicPlaylistID":144735,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8917.jpeg","UploadDate":"2017-03-21T08:25:04.4430000","DurationForVideoObject":"PT1M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"In this problem, we\u0027re told to find the density of a disc with a radius of"},{"Start":"00:03.525 ","End":"00:07.800","Text":"R and mass M. The disc then has a hole drilled out of it with a radius r,"},{"Start":"00:07.800 ","End":"00:10.610","Text":"and we need to find the mass that has been removed from the disc."},{"Start":"00:10.610 ","End":"00:13.365","Text":"To find the initial density of the large disc,"},{"Start":"00:13.365 ","End":"00:17.115","Text":"we need to do Sigma equals M mass over S area."},{"Start":"00:17.115 ","End":"00:19.080","Text":"Now we happen to know that the area"},{"Start":"00:19.080 ","End":"00:21.225","Text":"here is the area of a circle because we\u0027re looking at a disc,"},{"Start":"00:21.225 ","End":"00:24.720","Text":"so the formula for that is going to be the radius squared times Pi."},{"Start":"00:24.720 ","End":"00:28.035","Text":"We know that our area is Pi R^2,"},{"Start":"00:28.035 ","End":"00:30.120","Text":"our mass is M. Both these are given to us as"},{"Start":"00:30.120 ","End":"00:33.360","Text":"constant so that is our density of the disc,"},{"Start":"00:33.360 ","End":"00:35.310","Text":"M over Pi R squared."},{"Start":"00:35.310 ","End":"00:37.050","Text":"Now if there\u0027s a small hole drilled out,"},{"Start":"00:37.050 ","End":"00:39.045","Text":"as you see that we just drew in here,"},{"Start":"00:39.045 ","End":"00:42.105","Text":"we need to find the mass of the removed portion."},{"Start":"00:42.105 ","End":"00:45.470","Text":"The little mass m is going to be sigma times"},{"Start":"00:45.470 ","End":"00:50.645","Text":"s. That is the density times the area is going to give us the mass."},{"Start":"00:50.645 ","End":"00:53.075","Text":"Now, the density we have from before,"},{"Start":"00:53.075 ","End":"00:55.550","Text":"M over Pi R^2, R^2,"},{"Start":"00:55.550 ","End":"00:59.510","Text":"and we know that the area is Pi r squared because,"},{"Start":"00:59.510 ","End":"01:01.555","Text":"again, we\u0027re using the radius of a circle."},{"Start":"01:01.555 ","End":"01:04.660","Text":"Then once we simplify that,"},{"Start":"01:04.900 ","End":"01:11.840","Text":"bring it all together we have M times r over R squared and there you have it."},{"Start":"01:11.840 ","End":"01:16.830","Text":"That\u0027s the mass of the removed portion and the density of the large disc."}],"ID":9191}],"Thumbnail":null,"ID":144735},{"Name":"Infinitesimal Density","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Infintesimal Mass Element","Duration":"5m 3s","ChapterTopicVideoID":8918,"CourseChapterTopicPlaylistID":144736,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8918.jpeg","UploadDate":"2017-03-21T08:26:26.1670000","DurationForVideoObject":"PT5M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.530","Text":"In the previous lecture,"},{"Start":"00:01.530 ","End":"00:02.610","Text":"we talked about density."},{"Start":"00:02.610 ","End":"00:03.900","Text":"We talked about volume density,"},{"Start":"00:03.900 ","End":"00:06.820","Text":"area density, and linear density."},{"Start":"00:07.220 ","End":"00:10.650","Text":"What I want to do in this lecture is talk about a mass element."},{"Start":"00:10.650 ","End":"00:13.350","Text":"This is when we take a very infinitesimally small piece"},{"Start":"00:13.350 ","End":"00:16.575","Text":"of an object and investigate it in a different way."},{"Start":"00:16.575 ","End":"00:19.050","Text":"Just like we did before with our coordinates,"},{"Start":"00:19.050 ","End":"00:23.085","Text":"we\u0027re going to take an infinitesimally small object and we\u0027re going to name it dv."},{"Start":"00:23.085 ","End":"00:25.050","Text":"What we can do if we want to find"},{"Start":"00:25.050 ","End":"00:29.165","Text":"the density of it is we use the same formula from before,"},{"Start":"00:29.165 ","End":"00:31.145","Text":"except instead of it being m and v,"},{"Start":"00:31.145 ","End":"00:37.140","Text":"we\u0027re going to have dv and dm because both of those things are infinitesimally small."},{"Start":"00:37.880 ","End":"00:41.250","Text":"Instead of Rho=m/v,"},{"Start":"00:41.250 ","End":"00:46.950","Text":"we\u0027re going to have dm instead of m,"},{"Start":"00:46.950 ","End":"00:48.930","Text":"we\u0027re going to have Rho as the same,"},{"Start":"00:48.930 ","End":"00:51.645","Text":"and we\u0027re going to have dv instead of v. Why is there no d Rho?"},{"Start":"00:51.645 ","End":"00:53.180","Text":"Well, that\u0027s because Rho is a constant,"},{"Start":"00:53.180 ","End":"00:55.370","Text":"Rho is not going to be infinitesimally small."},{"Start":"00:55.370 ","End":"00:57.456","Text":"Rho is the same as before,"},{"Start":"00:57.456 ","End":"01:01.465","Text":"it\u0027s not affected by this procedure."},{"Start":"01:01.465 ","End":"01:05.015","Text":"What we\u0027re doing is taking a very small portion of our object,"},{"Start":"01:05.015 ","End":"01:06.650","Text":"a very small slice,"},{"Start":"01:06.650 ","End":"01:09.690","Text":"and we\u0027re treating it as though it\u0027s regular object."},{"Start":"01:09.690 ","End":"01:11.650","Text":"That\u0027s why we use dm and dv,"},{"Start":"01:11.650 ","End":"01:15.770","Text":"is because we\u0027re trying to remind ourselves that this is an infinitesimally small point."},{"Start":"01:15.770 ","End":"01:17.975","Text":"The reason we do this is actually,"},{"Start":"01:17.975 ","End":"01:19.160","Text":"for a very specific case,"},{"Start":"01:19.160 ","End":"01:20.855","Text":"it\u0027s very useful for us."},{"Start":"01:20.855 ","End":"01:23.900","Text":"If you have inconsistent density,"},{"Start":"01:23.900 ","End":"01:25.790","Text":"if you recall, we can\u0027t use our normal formula."},{"Start":"01:25.790 ","End":"01:28.010","Text":"We can\u0027t say Rho=m/v."},{"Start":"01:28.010 ","End":"01:30.290","Text":"We can\u0027t use any of the other versions of that function,"},{"Start":"01:30.290 ","End":"01:34.490","Text":"a formula such as m=Rho times v. Let\u0027s say for example,"},{"Start":"01:34.490 ","End":"01:38.090","Text":"that our density is inconsistent and it\u0027s dependent on"},{"Start":"01:38.090 ","End":"01:42.845","Text":"different parts of the object where you are dictates your density."},{"Start":"01:42.845 ","End":"01:46.445","Text":"In this case, we have Rho as a function of x, y, and z."},{"Start":"01:46.445 ","End":"01:49.730","Text":"In 1 place, you might have a high density in 1 place,"},{"Start":"01:49.730 ","End":"01:52.130","Text":"a low density in another place, a medium density."},{"Start":"01:52.130 ","End":"01:53.929","Text":"Let\u0027s say for example,"},{"Start":"01:53.929 ","End":"01:59.840","Text":"that Rho 0 is equal to x times y over a times b,"},{"Start":"01:59.840 ","End":"02:04.385","Text":"which are constants, plus z/c."},{"Start":"02:04.385 ","End":"02:07.160","Text":"Now, this is not a constant density,"},{"Start":"02:07.160 ","End":"02:09.140","Text":"it depends on where you are within the objects."},{"Start":"02:09.140 ","End":"02:13.635","Text":"You can\u0027t use that normal function and you can\u0027t use a normal formula."},{"Start":"02:13.635 ","End":"02:24.555","Text":"If I were to write out m=Rho (x,y,z) times v,"},{"Start":"02:24.555 ","End":"02:27.750","Text":"it wouldn\u0027t work because I\u0027m using something that\u0027s not constant,"},{"Start":"02:27.750 ","End":"02:29.075","Text":"I\u0027m using something that\u0027s variable."},{"Start":"02:29.075 ","End":"02:31.910","Text":"But if I\u0027m using dm and dv,"},{"Start":"02:31.910 ","End":"02:33.125","Text":"I\u0027m talking about a little point,"},{"Start":"02:33.125 ","End":"02:34.220","Text":"an infinitesimally small point,"},{"Start":"02:34.220 ","End":"02:36.785","Text":"so I can assume that in that particular point,"},{"Start":"02:36.785 ","End":"02:38.060","Text":"if I know my x, my y,"},{"Start":"02:38.060 ","End":"02:40.460","Text":"and my z coordinates if we\u0027re using Cartesian coordinates,"},{"Start":"02:40.460 ","End":"02:45.205","Text":"I have a consistent density in that exact point."},{"Start":"02:45.205 ","End":"02:48.530","Text":"What\u0027s useful about this is in this infinitesimally small point,"},{"Start":"02:48.530 ","End":"02:51.350","Text":"our density cannot change much."},{"Start":"02:51.350 ","End":"02:56.015","Text":"It\u0027s close enough to consistent that we can then go ahead and use the prior formula."},{"Start":"02:56.015 ","End":"02:58.850","Text":"If we\u0027re using dm and we\u0027re using dv,"},{"Start":"02:58.850 ","End":"03:03.980","Text":"we can in fact say dm=Rho sub 0,"},{"Start":"03:03.980 ","End":"03:05.300","Text":"which is the Rho that we\u0027re using here,"},{"Start":"03:05.300 ","End":"03:09.065","Text":"the formula that we just laid out, times dv."},{"Start":"03:09.065 ","End":"03:13.610","Text":"In our case, considering that it\u0027s a cube and we\u0027ve used x, y, and z before,"},{"Start":"03:13.610 ","End":"03:15.140","Text":"probably using Cartesian coordinates,"},{"Start":"03:15.140 ","End":"03:18.440","Text":"so it\u0027s Rho x,y,z times dv."},{"Start":"03:18.440 ","End":"03:21.050","Text":"Again, the formula is, for our example,"},{"Start":"03:21.050 ","End":"03:26.540","Text":"dm=Rho x times y over a times b plus z/c times dv."},{"Start":"03:26.540 ","End":"03:28.655","Text":"Again, because we\u0027re in Cartesian coordinates,"},{"Start":"03:28.655 ","End":"03:30.860","Text":"or we can assume we are at this point using x,"},{"Start":"03:30.860 ","End":"03:32.495","Text":"y, and z and using a cube,"},{"Start":"03:32.495 ","End":"03:35.104","Text":"we can assume that our dv,"},{"Start":"03:35.104 ","End":"03:40.280","Text":"if you recall from before, is going to equal dx times dy times dz."},{"Start":"03:40.280 ","End":"03:44.450","Text":"With this formula, we can then plug in"},{"Start":"03:44.450 ","End":"03:49.090","Text":"our points and then we can find out exactly what our density is."},{"Start":"03:49.090 ","End":"03:51.460","Text":"Once you find the mass of this 1 point,"},{"Start":"03:51.460 ","End":"03:55.310","Text":"you can then do an integral of it with limits set to the size of your object."},{"Start":"03:55.310 ","End":"03:58.910","Text":"That\u0027ll basically in the same way that we were finding volume before,"},{"Start":"03:58.910 ","End":"04:03.740","Text":"will set you up to get the mass of the entire object."},{"Start":"04:03.740 ","End":"04:06.260","Text":"That way, even if your density is inconsistent,"},{"Start":"04:06.260 ","End":"04:09.320","Text":"you can find the mass of entire object in this method."},{"Start":"04:09.320 ","End":"04:12.425","Text":"Now, because we have 3 variables here,"},{"Start":"04:12.425 ","End":"04:15.155","Text":"we\u0027re going to have to do a triple integral."},{"Start":"04:15.155 ","End":"04:18.980","Text":"We\u0027ll talk about that in the exercises that follow."},{"Start":"04:18.980 ","End":"04:21.440","Text":"But for now, this is more or less what it looks"},{"Start":"04:21.440 ","End":"04:24.340","Text":"like and we\u0027ll show you how to do that in a moment."},{"Start":"04:24.340 ","End":"04:27.890","Text":"In short, this is the entire idea behind dm"},{"Start":"04:27.890 ","End":"04:30.980","Text":"and the purpose of it is that you can then find the mass of"},{"Start":"04:30.980 ","End":"04:34.970","Text":"a large object using integrals of"},{"Start":"04:34.970 ","End":"04:39.315","Text":"different dm points when you have an inconsistent density."},{"Start":"04:39.315 ","End":"04:42.275","Text":"When you don\u0027t have a constant density and you can\u0027t use your normal formulas,"},{"Start":"04:42.275 ","End":"04:46.130","Text":"you can then rely on this formula to find your overall mass."},{"Start":"04:46.130 ","End":"04:51.800","Text":"Of course, there\u0027s a slight change of using cylindrical or spherical coordinates,"},{"Start":"04:51.800 ","End":"04:54.035","Text":"but it\u0027s the same principle throughout."},{"Start":"04:54.035 ","End":"04:56.000","Text":"Just remember that if you take integrals and"},{"Start":"04:56.000 ","End":"04:58.640","Text":"all 3 dimensions of a given point to cover an entire object,"},{"Start":"04:58.640 ","End":"05:00.695","Text":"you can then find the overall mass of an object,"},{"Start":"05:00.695 ","End":"05:03.570","Text":"even if it has inconsistent density."}],"ID":9192},{"Watched":false,"Name":"Exercise - Stake with Non-Uniform Density","Duration":"1m 56s","ChapterTopicVideoID":8919,"CourseChapterTopicPlaylistID":144736,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8919.jpeg","UploadDate":"2017-03-21T08:26:36.9330000","DurationForVideoObject":"PT1M56S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.800","Text":"Hello. In this exercise,"},{"Start":"00:01.800 ","End":"00:04.155","Text":"we\u0027re talking about a stake with a non-uniform density."},{"Start":"00:04.155 ","End":"00:07.050","Text":"We need to calculate the total mass of a stake with a length L and"},{"Start":"00:07.050 ","End":"00:10.590","Text":"a mass density of Lambda as a function of x equals Lambda sub 0,"},{"Start":"00:10.590 ","End":"00:13.380","Text":"x/L, when x is the distance from the left edge"},{"Start":"00:13.380 ","End":"00:16.320","Text":"of the stake and L and Lambda are given constants."},{"Start":"00:16.320 ","End":"00:17.820","Text":"But that means we have a stake here."},{"Start":"00:17.820 ","End":"00:19.200","Text":"The further we move to the right,"},{"Start":"00:19.200 ","End":"00:22.205","Text":"the more the density changes as relation to x."},{"Start":"00:22.205 ","End":"00:24.405","Text":"If that\u0027s 1 x unit away,"},{"Start":"00:24.405 ","End":"00:26.670","Text":"then that density is different than another portion"},{"Start":"00:26.670 ","End":"00:29.865","Text":"over there that\u0027s going to be maybe 2 x or 3 x units away."},{"Start":"00:29.865 ","End":"00:31.530","Text":"To find the total mass,"},{"Start":"00:31.530 ","End":"00:33.675","Text":"what we\u0027re going to do is take a small point dm,"},{"Start":"00:33.675 ","End":"00:36.330","Text":"and we\u0027re going to do an integral on it to find the entire mass of"},{"Start":"00:36.330 ","End":"00:39.570","Text":"the object of the stake in this case."},{"Start":"00:39.570 ","End":"00:43.550","Text":"Because we\u0027re doing a length density or linear density,"},{"Start":"00:43.550 ","End":"00:46.820","Text":"we know that we want to do Lambda times dl,"},{"Start":"00:46.820 ","End":"00:50.588","Text":"dl being the small difference in length."},{"Start":"00:50.588 ","End":"00:52.515","Text":"Because we\u0027re doing a dm,"},{"Start":"00:52.515 ","End":"00:54.135","Text":"we can use this formula."},{"Start":"00:54.135 ","End":"00:57.305","Text":"Now, it\u0027s easier to say that this is on the x-axis."},{"Start":"00:57.305 ","End":"01:02.500","Text":"That way, we can set dl=dx for the sake of our equation."},{"Start":"01:02.500 ","End":"01:05.795","Text":"We can now plug in our Lambda based in the equation above."},{"Start":"01:05.795 ","End":"01:09.200","Text":"Lambda sub 0, x/l times dx."},{"Start":"01:09.200 ","End":"01:14.075","Text":"Now, we\u0027re doing integral on this and we need to find the limits."},{"Start":"01:14.075 ","End":"01:16.580","Text":"Now remember, we\u0027re measuring dx in terms of x,"},{"Start":"01:16.580 ","End":"01:20.720","Text":"not in terms of Lambda or L because we\u0027re talking about the x-axis,"},{"Start":"01:20.720 ","End":"01:26.905","Text":"so the stake goes from a length of 0 at the beginning to a length of L at the end."},{"Start":"01:26.905 ","End":"01:29.915","Text":"If we now know that our limit is from 0 to L,"},{"Start":"01:29.915 ","End":"01:31.948","Text":"we can perform our integral,"},{"Start":"01:31.948 ","End":"01:37.460","Text":"and what we end up with is Lambda sub 0 over L on the outside, x^2/2,"},{"Start":"01:37.460 ","End":"01:41.260","Text":"still needing to be plugged in with its limit, 0 to L,"},{"Start":"01:41.260 ","End":"01:43.445","Text":"and once we plug in that limit,"},{"Start":"01:43.445 ","End":"01:47.900","Text":"we find x sub 0 L^2/2L, and let me simplify that."},{"Start":"01:47.900 ","End":"01:52.475","Text":"Our final result is Lambda sub 0 L/2 and you\u0027ve solved it."},{"Start":"01:52.475 ","End":"01:55.770","Text":"That is your total mass of the stake."}],"ID":9193}],"Thumbnail":null,"ID":144736},{"Name":"Infinitesimal Calculus","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Implicit Derivative","Duration":"6m 36s","ChapterTopicVideoID":8920,"CourseChapterTopicPlaylistID":144737,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8920.jpeg","UploadDate":"2017-03-21T08:27:39.7670000","DurationForVideoObject":"PT6M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this exercise, we\u0027re given the function f( x,"},{"Start":"00:03.720 ","End":"00:11.145","Text":"y) equals y to the power of Sine x plus 6y plus e to the power of x^2 plus y^2 equals 0."},{"Start":"00:11.145 ","End":"00:14.670","Text":"If it equals 0, it always equals 0. It\u0027s important to remember."},{"Start":"00:14.670 ","End":"00:16.860","Text":"We need to find dy over dx,"},{"Start":"00:16.860 ","End":"00:19.395","Text":"and the way we\u0027re going to do that is do a full differential"},{"Start":"00:19.395 ","End":"00:22.730","Text":"of f. The best way to do that,"},{"Start":"00:22.730 ","End":"00:24.785","Text":"let\u0027s do a partial derivative."},{"Start":"00:24.785 ","End":"00:29.000","Text":"It\u0027s going to be symbolized by a curly d. We\u0027re going to call it partial d. Partial"},{"Start":"00:29.000 ","End":"00:33.170","Text":"d of f over dx times dx. Now what does that mean?"},{"Start":"00:33.170 ","End":"00:35.450","Text":"It means that we\u0027re only doing a derivative"},{"Start":"00:35.450 ","End":"00:38.225","Text":"of x and we\u0027re dealing with y as though it were a constant."},{"Start":"00:38.225 ","End":"00:41.450","Text":"Once we\u0027ve done that, we need to do the same thing for y,"},{"Start":"00:41.450 ","End":"00:46.680","Text":"partial derivative of f over dy times dy."},{"Start":"00:46.840 ","End":"00:49.880","Text":"Now of course this is going to be equal to 0"},{"Start":"00:49.880 ","End":"00:52.610","Text":"because our function above is also equal to 0."},{"Start":"00:52.610 ","End":"00:56.795","Text":"The best way to solve this from here is to separate our dy and our dx."},{"Start":"00:56.795 ","End":"01:00.770","Text":"We\u0027re going to take our dx and subtract it and put it over on the other side,"},{"Start":"01:00.770 ","End":"01:07.110","Text":"so we\u0027ll have partial df over dy multiplied by dy,"},{"Start":"01:07.110 ","End":"01:10.210","Text":"let me just write that out for a second,"},{"Start":"01:10.820 ","End":"01:21.675","Text":"equals negative partial df over dx times dx."},{"Start":"01:21.675 ","End":"01:24.720","Text":"In order to end up with dy over dx,"},{"Start":"01:24.720 ","End":"01:30.025","Text":"what I want to do is divide both sides by partial df over dy,"},{"Start":"01:30.025 ","End":"01:32.960","Text":"and then divide both sides by dx."},{"Start":"01:32.960 ","End":"01:35.090","Text":"I end up on my left side with dy/dx,"},{"Start":"01:35.090 ","End":"01:36.830","Text":"which is what we\u0027re trying to find, and on the right,"},{"Start":"01:36.830 ","End":"01:41.010","Text":"I am left with a method partial df over dx."},{"Start":"01:41.010 ","End":"01:45.150","Text":"Of course is going to be negative over partial df over dy."},{"Start":"01:45.150 ","End":"01:50.405","Text":"What I can do is then I can do my partial derivative of f over x,"},{"Start":"01:50.405 ","End":"01:53.730","Text":"partial derivative of f over y. I can put one over the other,"},{"Start":"01:53.730 ","End":"01:57.540","Text":"make sure that it\u0027s negative and then I\u0027ll end up with my answer."},{"Start":"01:58.490 ","End":"02:01.820","Text":"In the end, what we\u0027re doing here isn\u0027t so complicated."},{"Start":"02:01.820 ","End":"02:03.530","Text":"We\u0027re finding a partial differential of x,"},{"Start":"02:03.530 ","End":"02:05.300","Text":"we\u0027re finding a partial differential of y,"},{"Start":"02:05.300 ","End":"02:08.300","Text":"and we\u0027re going to plug them back into our equation that exist here."},{"Start":"02:08.300 ","End":"02:11.270","Text":"The actual math behind it is going to take a little bit of writing,"},{"Start":"02:11.270 ","End":"02:12.470","Text":"so I\u0027ll put it on the side here."},{"Start":"02:12.470 ","End":"02:14.105","Text":"Just bear with me for a moment."},{"Start":"02:14.105 ","End":"02:19.110","Text":"We\u0027re going to start clean. Let\u0027s write it out as we had."},{"Start":"02:19.110 ","End":"02:22.555","Text":"Y to the power of Sine x."},{"Start":"02:22.555 ","End":"02:27.425","Text":"We\u0027re going to need to find the derivative of that in relation to x."},{"Start":"02:27.425 ","End":"02:30.830","Text":"Write that out with an apostrophe."},{"Start":"02:30.830 ","End":"02:33.325","Text":"Just easier notation."},{"Start":"02:33.325 ","End":"02:35.870","Text":"We\u0027re going to use a little trick to change the form of"},{"Start":"02:35.870 ","End":"02:38.135","Text":"this so it\u0027s easier to do a derivative."},{"Start":"02:38.135 ","End":"02:40.700","Text":"We\u0027re going to do e to the power of ln,"},{"Start":"02:40.700 ","End":"02:47.450","Text":"the natural log, times y to the power of Sine x."},{"Start":"02:47.450 ","End":"02:51.920","Text":"You can do this with any equation because it continues to equal the initial equation,"},{"Start":"02:51.920 ","End":"02:55.025","Text":"e to the power of ln times your initial equation."},{"Start":"02:55.025 ","End":"02:57.365","Text":"We\u0027re going to get the derivative of that in relation to x."},{"Start":"02:57.365 ","End":"03:00.140","Text":"What this trick allows me to do is take the exponent,"},{"Start":"03:00.140 ","End":"03:04.360","Text":"in this case Sine x and bring it down to be multiplied by Ln."},{"Start":"03:04.360 ","End":"03:06.155","Text":"We can rewrite this equation,"},{"Start":"03:06.155 ","End":"03:12.570","Text":"still equal the same thing as e to the power of Sine x times ln of y."},{"Start":"03:12.570 ","End":"03:15.140","Text":"Of course we still haven\u0027t done a derivative,"},{"Start":"03:15.140 ","End":"03:18.100","Text":"so we need to remember to write in that symbol."},{"Start":"03:18.100 ","End":"03:21.470","Text":"What we\u0027ve done is now we have Sine x or variable"},{"Start":"03:21.470 ","End":"03:24.065","Text":"multiplied by some constant and the 2 are separated."},{"Start":"03:24.065 ","End":"03:25.850","Text":"We can do our derivative."},{"Start":"03:25.850 ","End":"03:29.425","Text":"When we do that, it\u0027s going to come out as follows,"},{"Start":"03:29.425 ","End":"03:37.780","Text":"e to the power of Sine x times lny multiplied by Iny or constant has to be multiplied,"},{"Start":"03:38.060 ","End":"03:43.385","Text":"and then we\u0027re going to also add in the internal derivative of Sine x,"},{"Start":"03:43.385 ","End":"03:46.905","Text":"the antiderivative Cosine x."},{"Start":"03:46.905 ","End":"03:50.390","Text":"What we can do now is we have something that\u0027s equal to our initial equation."},{"Start":"03:50.390 ","End":"03:53.075","Text":"We\u0027ve done the derivative and once we clean it up a little bit,"},{"Start":"03:53.075 ","End":"03:58.280","Text":"simplify it, we can bring it down below and we\u0027re halfway done with our solution."},{"Start":"03:58.280 ","End":"04:03.680","Text":"The more simplified version of this is lny times"},{"Start":"04:03.680 ","End":"04:12.360","Text":"Cosine x times y to the power of Sine x."},{"Start":"04:12.530 ","End":"04:17.210","Text":"This is the derivative of y to the power of Sine x."},{"Start":"04:17.210 ","End":"04:19.910","Text":"We can copy it down here and we\u0027re finishing off"},{"Start":"04:19.910 ","End":"04:25.055","Text":"our derivative handling the e to the x^2 plus y^2 power."},{"Start":"04:25.055 ","End":"04:32.435","Text":"We can add in 2x times e to the power of x^2 plus y^2."},{"Start":"04:32.435 ","End":"04:35.735","Text":"That\u0027s the derivative of that. Of course 6y gives us 0."},{"Start":"04:35.735 ","End":"04:40.405","Text":"If we want to do the partial derivative in relation to y,"},{"Start":"04:40.405 ","End":"04:42.340","Text":"we can do the same trick we did with x."},{"Start":"04:42.340 ","End":"04:46.980","Text":"We end up with e to the power of Sine x times lny."},{"Start":"04:46.980 ","End":"04:52.855","Text":"But this time, Sine x is going to be our constant and lny is a variable."},{"Start":"04:52.855 ","End":"04:55.375","Text":"But again, we\u0027ve separated them so it\u0027s useful to us."},{"Start":"04:55.375 ","End":"04:58.265","Text":"This time we\u0027re going to do the derivative in relation to y."},{"Start":"04:58.265 ","End":"05:01.680","Text":"What we end up with is Sine x drops down as our constant,"},{"Start":"05:01.680 ","End":"05:12.470","Text":"and we have Sine x multiplied by e to the power of lny times 1 over y."},{"Start":"05:12.680 ","End":"05:16.270","Text":"Now if we simplify and finish off our derivative,"},{"Start":"05:16.270 ","End":"05:19.630","Text":"we can write partial df over dy."},{"Start":"05:19.630 ","End":"05:25.480","Text":"The first thing we need to do is simplify this term that we\u0027ve just come to."},{"Start":"05:26.030 ","End":"05:28.380","Text":"What we can do is,"},{"Start":"05:28.380 ","End":"05:30.835","Text":"I forgot the Sine x before. I\u0027m sorry."},{"Start":"05:30.835 ","End":"05:36.360","Text":"We need to write Sine x times 1"},{"Start":"05:36.360 ","End":"05:41.640","Text":"over y times y to the power of Sine x."},{"Start":"05:41.640 ","End":"05:45.850","Text":"If we simplify using the 1 over y and y to the power something,"},{"Start":"05:45.850 ","End":"05:55.930","Text":"we end up with partial df over dy equaling Sine x times y to the power of Sine x minus 1."},{"Start":"05:56.300 ","End":"06:02.795","Text":"If we continue on with the derivative 6y becomes 6 and e to the power of x^2 plus y^2."},{"Start":"06:02.795 ","End":"06:05.480","Text":"It\u0027s the same as what we did with x, this time in relation to y."},{"Start":"06:05.480 ","End":"06:10.570","Text":"It\u0027s 2y times e to the power of x^2 plus y^2."},{"Start":"06:10.570 ","End":"06:13.285","Text":"Now you\u0027ve solved partial df over dy."},{"Start":"06:13.285 ","End":"06:17.840","Text":"What you can do is take that solution and put it over partial df,"},{"Start":"06:17.840 ","End":"06:23.035","Text":"dx, add in your negative sign and you\u0027ve solved."},{"Start":"06:23.035 ","End":"06:24.620","Text":"If we just copy this in,"},{"Start":"06:24.620 ","End":"06:25.910","Text":"it\u0027s not going to be super simplified,"},{"Start":"06:25.910 ","End":"06:26.960","Text":"but this is our solution,"},{"Start":"06:26.960 ","End":"06:30.890","Text":"dy over dx equals negative what we"},{"Start":"06:30.890 ","End":"06:36.300","Text":"found for dy over what we found for dx. There you have it."}],"ID":9194},{"Watched":false,"Name":"Changing Coordinates of a Linear Element","Duration":"8m ","ChapterTopicVideoID":8921,"CourseChapterTopicPlaylistID":144737,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/8921.jpeg","UploadDate":"2017-03-21T08:28:25.8800000","DurationForVideoObject":"PT8M","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.794","Text":"Hello. In this exercise we\u0027re given 2 new coordinates;"},{"Start":"00:03.794 ","End":"00:05.280","Text":"r-tag and Theta-tag,"},{"Start":"00:05.280 ","End":"00:09.900","Text":"and they are replacing our initial cylindrical coordinates which are r and Theta."},{"Start":"00:09.900 ","End":"00:11.430","Text":"What we\u0027re supposed to do is find"},{"Start":"00:11.430 ","End":"00:16.229","Text":"the length element dl as a function of the new coordinates."},{"Start":"00:16.229 ","End":"00:20.700","Text":"What we need to do is on a basic level,"},{"Start":"00:20.700 ","End":"00:22.500","Text":"we can write out the formula for dl,"},{"Start":"00:22.500 ","End":"00:24.540","Text":"and we can actually start by writing this"},{"Start":"00:24.540 ","End":"00:26.590","Text":"as dl^2=dx^2"},{"Start":"00:33.020 ","End":"00:35.505","Text":"plus dy^2."},{"Start":"00:35.505 ","End":"00:36.810","Text":"If we\u0027re working in 3-dimensions,"},{"Start":"00:36.810 ","End":"00:38.845","Text":"we can also talk about dz^2."},{"Start":"00:38.845 ","End":"00:40.820","Text":"Right now we\u0027re only talking about 2 coordinates,"},{"Start":"00:40.820 ","End":"00:42.650","Text":"so we can assume we\u0027re in 2-dimensions."},{"Start":"00:42.650 ","End":"00:45.070","Text":"We\u0027ll get rid of our dz for now."},{"Start":"00:45.070 ","End":"00:50.450","Text":"These are my initial coordinates in Cartesian coordinates."},{"Start":"00:50.450 ","End":"00:53.795","Text":"If I want to transfer them into cylindrical coordinates,"},{"Start":"00:53.795 ","End":"00:57.325","Text":"what I need to do is remember some of the formulas we talked about before."},{"Start":"00:57.325 ","End":"01:02.700","Text":"First of all if you recall x=r cosine"},{"Start":"01:02.700 ","End":"01:08.163","Text":"Theta and y=r sine Theta,"},{"Start":"01:08.163 ","End":"01:11.600","Text":"and I\u0027m going to use that to convert this into cylindrical coordinates."},{"Start":"01:11.600 ","End":"01:15.065","Text":"What I do have is I don\u0027t have x or y, I have dx and dy."},{"Start":"01:15.065 ","End":"01:17.750","Text":"If I want to do dx, I want to do a differential here."},{"Start":"01:17.750 ","End":"01:19.351","Text":"I have 2 variables;"},{"Start":"01:19.351 ","End":"01:22.003","Text":"I have r and I have Theta,"},{"Start":"01:22.003 ","End":"01:24.940","Text":"and we can imagine that x is a function of r and Theta."},{"Start":"01:24.940 ","End":"01:30.770","Text":"I need to do partial differentials to get the answer that I\u0027m looking for,"},{"Start":"01:30.770 ","End":"01:34.090","Text":"so we\u0027re going to do partial differential x over"},{"Start":"01:34.090 ","End":"01:37.690","Text":"partial r times dr and then we\u0027re going to add to"},{"Start":"01:37.690 ","End":"01:48.430","Text":"that partial dx over partial dTheta times dTheta."},{"Start":"01:48.800 ","End":"01:53.020","Text":"First let\u0027s solve for x or dx really."},{"Start":"01:53.020 ","End":"01:56.305","Text":"If I do a partial differential for r,"},{"Start":"01:56.305 ","End":"02:01.699","Text":"the r drops out and I\u0027m left with cosine Theta dr."},{"Start":"02:01.699 ","End":"02:09.725","Text":"When I solve for Theta the plus turns into a minus,"},{"Start":"02:09.725 ","End":"02:15.530","Text":"and I\u0027m left with r sine Theta dTheta."},{"Start":"02:16.160 ","End":"02:19.944","Text":"Now let\u0027s find dy once we have dx."},{"Start":"02:19.944 ","End":"02:21.620","Text":"Dy is going to be the same idea."},{"Start":"02:21.620 ","End":"02:24.500","Text":"We have dy in terms of r,"},{"Start":"02:24.500 ","End":"02:27.995","Text":"n in terms of Theta, so we need to do partial derivatives."},{"Start":"02:27.995 ","End":"02:35.305","Text":"Partial dy over partial dr times dr plus partial dy over partial dTheta times dTheta,"},{"Start":"02:35.305 ","End":"02:39.305","Text":"and when we solve for that we end up with a very similar thing as you can imagine."},{"Start":"02:39.305 ","End":"02:46.950","Text":"We end up with sine Theta dr plus r cosine Theta dTheta."},{"Start":"02:47.240 ","End":"02:53.615","Text":"If I want to plug these back into my initial equation of dl^2=dx^2 plus dy^2,"},{"Start":"02:53.615 ","End":"02:57.155","Text":"I know that I need to square all of my elements here."},{"Start":"02:57.155 ","End":"03:00.560","Text":"Because we have 2 terms; 1 minus the other,"},{"Start":"03:00.560 ","End":"03:02.330","Text":"I don\u0027t need to go through the parentheses,"},{"Start":"03:02.330 ","End":"03:03.940","Text":"I can just simplify it right away."},{"Start":"03:03.940 ","End":"03:13.665","Text":"I\u0027m going to end up with this cosine^2 Theta dr^2 minus 2r cosine"},{"Start":"03:13.665 ","End":"03:18.570","Text":"Theta sine Theta dr dTheta"},{"Start":"03:18.570 ","End":"03:25.215","Text":"plus r^2 sine^2 Theta dTheta^2."},{"Start":"03:25.215 ","End":"03:31.080","Text":"That\u0027s going to be my x."},{"Start":"03:31.080 ","End":"03:40.815","Text":"For y, we have sine^2 Theta dr^2 plus"},{"Start":"03:40.815 ","End":"03:47.460","Text":"2r cosine Theta sine Theta dr dTheta"},{"Start":"03:47.460 ","End":"03:56.925","Text":"plus r^2 cosine^2 Theta dTheta^2."},{"Start":"03:56.925 ","End":"03:59.375","Text":"When we simplify this, our 2 middle terms drop out."},{"Start":"03:59.375 ","End":"04:01.865","Text":"We have minus 2 and plus 2 the same term."},{"Start":"04:01.865 ","End":"04:05.870","Text":"If we have cosine^2 Theta plus sine^2 Theta, that equals 1."},{"Start":"04:05.870 ","End":"04:11.300","Text":"So they drop out and we end up with dr^2."},{"Start":"04:11.300 ","End":"04:13.151","Text":"Plus on the other side we have the same thing;"},{"Start":"04:13.151 ","End":"04:15.500","Text":"cosine^2 Theta and sine^2 Theta,"},{"Start":"04:15.500 ","End":"04:21.810","Text":"and they drop out and we end up with r^2 dTheta^2."},{"Start":"04:21.830 ","End":"04:24.845","Text":"What you have here is your response,"},{"Start":"04:24.845 ","End":"04:29.642","Text":"your answer in terms of your initial cylindrical coordinates of Theta and r,"},{"Start":"04:29.642 ","End":"04:35.840","Text":"but we need to turn that into our new cylindrical coordinates of r-tag and Theta-tag."},{"Start":"04:35.840 ","End":"04:39.965","Text":"The way that we\u0027re going to do that is we\u0027re going to think of things"},{"Start":"04:39.965 ","End":"04:44.590","Text":"in terms of r and in terms of Theta relative to r-tag and Theta-tag."},{"Start":"04:44.590 ","End":"04:47.775","Text":"What we have is we have 2 formulas,"},{"Start":"04:47.775 ","End":"04:51.350","Text":"2 equations that give values for r-tag and Theta-tag."},{"Start":"04:51.350 ","End":"04:53.250","Text":"We don\u0027t have them in terms of r,"},{"Start":"04:53.250 ","End":"04:55.845","Text":"we don\u0027t have in terms of Theta, we have them in terms of our new coordinates."},{"Start":"04:55.845 ","End":"04:57.575","Text":"Let\u0027s think of them in terms of our old coordinates,"},{"Start":"04:57.575 ","End":"05:05.351","Text":"r. So Theta-tag and r-tag is a function of r. We can invert the r-tag formula."},{"Start":"05:05.351 ","End":"05:10.850","Text":"Right now r-tag=1 over r^2."},{"Start":"05:10.850 ","End":"05:21.940","Text":"Instead, it can be written as r=1 over r tag^1/2 or are tag^negative 1/2."},{"Start":"05:21.940 ","End":"05:26.630","Text":"For Theta we can just reverse the formula again,"},{"Start":"05:26.630 ","End":"05:31.030","Text":"multiply everything by 2, and we have Theta=Theta-tag."},{"Start":"05:31.030 ","End":"05:33.980","Text":"Now that I have this in terms of r and Theta,"},{"Start":"05:33.980 ","End":"05:37.100","Text":"I can take my partial differentials, my partial derivatives."},{"Start":"05:37.100 ","End":"05:43.040","Text":"Dr=partial dr over partial dr-tag times"},{"Start":"05:43.040 ","End":"05:51.320","Text":"dr-tag plus partial dr over partial dTheta-tag times dTheta-tag."},{"Start":"05:51.320 ","End":"05:53.165","Text":"All this in parentheses of course."},{"Start":"05:53.165 ","End":"05:57.740","Text":"In this equation above r is not in terms of Theta at all,"},{"Start":"05:57.740 ","End":"06:00.360","Text":"so that will go to 0."},{"Start":"06:01.400 ","End":"06:06.785","Text":"What we have then is our partial dr over partial dr tag."},{"Start":"06:06.785 ","End":"06:08.390","Text":"When we do the derivative there,"},{"Start":"06:08.390 ","End":"06:17.615","Text":"we end up with negative 1/2r-tag^negative 3/2dr-tag."},{"Start":"06:17.615 ","End":"06:23.195","Text":"You can do the same thing for Theta and solve for your differential of Theta."},{"Start":"06:23.195 ","End":"06:28.830","Text":"We can do a partial derivative Theta over partial dr-tag times"},{"Start":"06:28.830 ","End":"06:35.795","Text":"dr-tag plus partial dTheta over partial dTheta-tag times dTheta tag."},{"Start":"06:35.795 ","End":"06:41.690","Text":"Once again we\u0027re in a situation where Theta is not defined in terms of r-tag,"},{"Start":"06:41.690 ","End":"06:43.375","Text":"so that will turn to 0."},{"Start":"06:43.375 ","End":"06:47.255","Text":"When we solve for partial dTheta over partial dTheta tag,"},{"Start":"06:47.255 ","End":"06:50.960","Text":"we end up with 2dTheta-tag."},{"Start":"06:50.960 ","End":"06:55.280","Text":"If I want to plug that into our equation from earlier,"},{"Start":"06:55.280 ","End":"07:04.850","Text":"remember dl^2=dx^2 plus dy^2 which also equals dr^2 plus r^2 dTheta^2."},{"Start":"07:04.850 ","End":"07:09.250","Text":"So I can take the terms that we have now and plug them in for r instead."},{"Start":"07:09.250 ","End":"07:13.750","Text":"Instead of dr^2 I\u0027m going to put in"},{"Start":"07:13.940 ","End":"07:20.805","Text":"1/4r-tag^negative 3 times dr tag^2,"},{"Start":"07:20.805 ","End":"07:24.180","Text":"and that\u0027s just if you do the math on dr^2,"},{"Start":"07:24.180 ","End":"07:29.311","Text":"what you end up with and I\u0027m going to add to that."},{"Start":"07:29.311 ","End":"07:33.170","Text":"Instead of r squared I don\u0027t have to actually use"},{"Start":"07:33.170 ","End":"07:37.370","Text":"the dr as I can go back to our initial equation and"},{"Start":"07:37.370 ","End":"07:47.520","Text":"say that if r=r-tag^1/2 then r^2=1 over r-tag,"},{"Start":"07:47.520 ","End":"07:53.400","Text":"and multiply that by dTheta^2 which ends up as 4dTheta tag^2."},{"Start":"07:53.400 ","End":"07:55.890","Text":"If I want the value of dl instead of"},{"Start":"07:55.890 ","End":"07:58.760","Text":"dl^2 I just take the square root of this answer I\u0027ve got,"},{"Start":"07:58.760 ","End":"08:00.990","Text":"and now I have my solution."}],"ID":9195}],"Thumbnail":null,"ID":144737}]

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