1 00:00:00,450 --> 00:00:06,540 So let us now talk about different coordinate systems, and first of all, I want to introduce to you 2 00:00:06,750 --> 00:00:12,300 the Cartesian coordinate system, which is the standard coordinate system that all of you know. 3 00:00:13,500 --> 00:00:21,210 So in Cartesian coordinates, you have a position vector that is characterized by three components, 4 00:00:21,450 --> 00:00:22,810 X, Y and Z. 5 00:00:23,760 --> 00:00:30,960 So if you introduce these units vectors here that point along these directions and have the length of 6 00:00:30,960 --> 00:00:39,050 one, then you can write your position vector as X times X plus Y times E Y plus three times easy. 7 00:00:39,660 --> 00:00:45,840 And this gives you the typical vector notation X, Y, Z for the three components. 8 00:00:47,330 --> 00:00:50,340 Now, here you can see that everything is really uniform. 9 00:00:50,450 --> 00:00:57,620 You have these lines, these grid lines, and the grid looks the same and along every direction at every 10 00:00:57,620 --> 00:00:58,680 point in space. 11 00:00:59,240 --> 00:01:00,410 So it's really uniform. 12 00:01:01,040 --> 00:01:05,750 And this leads to very simple line elements for integration. 13 00:01:06,080 --> 00:01:11,870 Very simple surface elements for integration and very simple volume elements for integration. 14 00:01:12,380 --> 00:01:20,570 So you can really separate your line and three dimensional space and equidistant elements and you can 15 00:01:20,570 --> 00:01:26,120 separate, for example, the volume in cubes that have the same size. 16 00:01:26,120 --> 00:01:31,130 And they are characterized by the length deep divide and easy. 17 00:01:31,910 --> 00:01:36,920 So of course, for many problems, it's very, very useful to consider Cartesian coordinates. 18 00:01:37,520 --> 00:01:42,890 So if you want to do some integration in three dimensional space, you do not have to think about what 19 00:01:42,890 --> 00:01:44,450 the volume element looks like. 20 00:01:44,450 --> 00:01:47,330 You just write it down like this and you're good to go. 21 00:01:48,230 --> 00:01:54,530 However, there are, of course, other problems where it is not useful and a very famous one is when 22 00:01:54,530 --> 00:01:57,390 you try to calculate the volume of a sphere. 23 00:01:58,940 --> 00:02:06,050 So in this case, if you consider such a cubic volume elements where you introduce many, many cubes 24 00:02:06,050 --> 00:02:12,410 that are very small in size and you try to fill up this cube, there's the sphere. 25 00:02:12,410 --> 00:02:18,470 I'm sorry, then it's very difficult to do so because what you would have to do is you would have to 26 00:02:18,470 --> 00:02:22,010 parametrized a sphere in Cartesian coordinates. 27 00:02:22,370 --> 00:02:27,730 What this means is you must have an equation for the surface of the sphere. 28 00:02:28,100 --> 00:02:33,470 You have to know how to express this in terms of the X, Y and Z coordinate. 29 00:02:34,250 --> 00:02:39,920 So you get some square roots and everything is quite difficult and you have to be very, very careful 30 00:02:40,070 --> 00:02:42,900 about what you are doing and you have to think very carefully. 31 00:02:43,850 --> 00:02:50,900 So in this case, it's probably not that useful to consider such cubic volume elements, and it's much 32 00:02:50,900 --> 00:02:58,580 better to use volume elements that already are in such a circular shape and already account for the 33 00:02:58,580 --> 00:03:01,010 sphere in terms of the geometry. 34 00:03:01,940 --> 00:03:06,680 So for such problems, we can use alternative coordinate systems. 35 00:03:07,340 --> 00:03:15,590 And next, I want to start with the with the polar coordinates, which is one of the most used coordinate 36 00:03:15,590 --> 00:03:18,490 systems, but it's only in two dimensional space. 37 00:03:18,770 --> 00:03:22,300 And later on, I will generalize this in three dimensional space. 38 00:03:23,270 --> 00:03:30,610 So in polar coordinates, we consider such a, um, circular coordinate system. 39 00:03:31,190 --> 00:03:39,810 So instead of considering these coordinates, X and Y, we consider the length of our vector R and a 40 00:03:39,890 --> 00:03:41,290 polar angle five. 41 00:03:41,780 --> 00:03:44,000 You can see this is the polar angle. 42 00:03:44,170 --> 00:03:45,870 Fine, this is the vector. 43 00:03:46,250 --> 00:03:53,300 And the length of the vector is, of course, are when you do not write it as a vector is just a scalar, 44 00:03:53,300 --> 00:03:54,440 it's not a written bolt. 45 00:03:55,250 --> 00:03:57,200 And there's of course a transformation. 46 00:03:57,470 --> 00:04:04,580 You can calculate the length of the vector R by calculating the square root of the X component squared 47 00:04:04,580 --> 00:04:06,020 plus the Y component square. 48 00:04:06,710 --> 00:04:13,730 So this is just because you have a rectangular triangle and you can calculate the and the polar angle 49 00:04:13,730 --> 00:04:16,470 fine by using the Yarkas tangency function. 50 00:04:17,300 --> 00:04:22,790 So typically what you find is you calculate the arcus tongue and function of the Y component divided 51 00:04:22,790 --> 00:04:23,870 by the X component. 52 00:04:24,350 --> 00:04:29,660 But that's not uniquely defined because it's very easy to think about it when your point is. 53 00:04:29,660 --> 00:04:33,010 For example, somewhere here you have a positive X and Y. 54 00:04:33,380 --> 00:04:39,290 So this ratio here is positive and when your point is here, you have negative X and Y, and so this 55 00:04:39,290 --> 00:04:40,730 ratio is also positive. 56 00:04:40,730 --> 00:04:43,210 It would give you the same angle by calculating this. 57 00:04:43,820 --> 00:04:47,350 So you really have to account also for the sign of X and Y. 58 00:04:47,720 --> 00:04:54,570 And in most programming languages you have for this, um, a predefined function that is the second 59 00:04:54,590 --> 00:04:55,840 across its function. 60 00:04:55,850 --> 00:05:02,750 It's most of the time it's written eight, 10, two, and all it does is it takes the result from the 61 00:05:02,750 --> 00:05:09,950 Archos Tongans and for some points it adds another PI to account for this sign. 62 00:05:11,270 --> 00:05:17,240 OK, so we can write down in this coordinate system our position vector in terms of unit vectors as 63 00:05:17,240 --> 00:05:17,570 well. 64 00:05:18,260 --> 00:05:20,330 However, here it's a bit different. 65 00:05:20,460 --> 00:05:23,530 You do not add up three different unit vectors. 66 00:05:23,540 --> 00:05:25,910 So here it's really just a single one. 67 00:05:26,630 --> 00:05:33,110 This is because if you look at these unit vectors, you can see you have the E.R. unit vector pointing 68 00:05:33,110 --> 00:05:34,480 along the R direction. 69 00:05:35,150 --> 00:05:40,100 So if you want to express this vector R, then you do not need this vector here. 70 00:05:40,100 --> 00:05:41,450 You only need this one. 71 00:05:41,990 --> 00:05:46,190 You have to take this vector which has the length one and then you want to. 72 00:05:46,480 --> 00:05:54,970 By the length of the vector art itself, and that's the whole trick, you really have already accounted 73 00:05:54,970 --> 00:05:56,110 for the geometry. 74 00:05:56,500 --> 00:05:59,780 So, for example, you have a function that only depends on the radius. 75 00:06:00,280 --> 00:06:06,910 So then it's much better to use this coordinate system because you only need to use one vector to express 76 00:06:06,910 --> 00:06:07,860 the position vector. 77 00:06:07,900 --> 00:06:09,460 That's already a great advantage. 78 00:06:10,750 --> 00:06:14,380 Now, of course, you also, for some problems, need the five vector. 79 00:06:14,710 --> 00:06:20,680 For example, if you want to calculate the change of our velocity, then it can of course happen that 80 00:06:20,680 --> 00:06:24,130 it has some transverse component where you need to see five vector. 81 00:06:24,520 --> 00:06:27,460 So this is also very much needed. 82 00:06:28,330 --> 00:06:34,660 And like in Cartesian coordinates, if you calculate the scale of product of these two vectors, it 83 00:06:34,660 --> 00:06:37,390 gives zero, which means they are perpendicular. 84 00:06:37,630 --> 00:06:39,490 This is because of this minus sign here. 85 00:06:41,350 --> 00:06:47,470 So this means we can just take this vector, multiply the length, and this allows us to express the 86 00:06:47,470 --> 00:06:51,190 position are in terms of this two component vector. 87 00:06:53,250 --> 00:07:00,900 Now, this was just where the polar coordinates in two dimensions and the three dimensional generalization 88 00:07:00,900 --> 00:07:07,980 of this coordinate system is called the cylindrical coordinates, because here what we do is we take 89 00:07:07,980 --> 00:07:16,470 our polar coordinates from 2D, where we had only the X and Y axis, and we just add a uniform component 90 00:07:16,500 --> 00:07:16,880 Z. 91 00:07:17,220 --> 00:07:30,930 So this means we extrude this circle to a cylinder and transformation's for X and Y and the length in 92 00:07:30,930 --> 00:07:37,590 the polar plane, which is this time called row and depolarizing of it's exactly the same as on the 93 00:07:37,590 --> 00:07:38,490 previous slide. 94 00:07:38,910 --> 00:07:43,170 So if you know about polar coordinates, you also know about cylindrical coordinates. 95 00:07:44,370 --> 00:07:50,400 And you can see since we have this this uniformity along this direction, we do not have to transform 96 00:07:50,400 --> 00:07:51,370 this lead component. 97 00:07:51,390 --> 00:07:57,930 It will stay the same in Cartesian coordinates and in these cylindrical coordinates. 98 00:07:58,800 --> 00:08:05,940 So these coordinates are, for example, useful if you have a wire going along the Z direction and then 99 00:08:05,940 --> 00:08:13,740 you want to calculate, for example, the magnetic field that will look the same for every Z component, 100 00:08:13,740 --> 00:08:15,930 for every cut through the cylinder. 101 00:08:16,230 --> 00:08:21,230 And then you just have to look what it looks like in this X Y plane. 102 00:08:21,780 --> 00:08:24,500 So therefore you can use cylindrical coordinates. 103 00:08:25,770 --> 00:08:35,380 And one important comment is that here we introduce a length rule, which is not the length of the vector, 104 00:08:35,400 --> 00:08:42,390 are it is the length of the vector if you take this vector and projected onto the X Y plane. 105 00:08:43,740 --> 00:08:49,770 So this is why I did not call this are here because we are in three dimensions would be X squared plus 106 00:08:49,770 --> 00:08:55,540 Y squared plus the square and then the square root that we do not account for the Z component here. 107 00:08:57,390 --> 00:09:02,520 So the position vector in this case can be expressed like in polar coordinates. 108 00:09:02,520 --> 00:09:07,710 We have the length in this polar plane times the unit vector in this polar plane. 109 00:09:08,460 --> 00:09:10,860 But then we also need the Z component. 110 00:09:10,950 --> 00:09:12,720 This is why we add here the second term. 111 00:09:13,650 --> 00:09:16,230 And when you look now at the unit vectors, they look like this. 112 00:09:16,680 --> 00:09:25,190 So you have a very real vector which would point along this right direction here. 113 00:09:25,380 --> 00:09:29,010 So it would just extrude this line here, this dashed line. 114 00:09:30,180 --> 00:09:37,740 So it points along the direction of the vector R if you take it and project it into the X Y plane and 115 00:09:37,740 --> 00:09:41,140 then it has the same shape as previously. 116 00:09:41,580 --> 00:09:43,600 So yeah, here's a small mistake here. 117 00:09:43,620 --> 00:09:51,550 This must be zero and not e r then we have the EFI vector which is also the same like in polar coordinates. 118 00:09:51,930 --> 00:10:01,110 You could put this vector also here and would point um, tangentially along this, this gray line here. 119 00:10:02,340 --> 00:10:06,720 And then we have to Z unit vector, which is the same as in Cartesian coordinates. 120 00:10:07,020 --> 00:10:09,120 It just points along in this direction. 121 00:10:11,040 --> 00:10:11,610 All right. 122 00:10:11,640 --> 00:10:17,660 So now you may be wondering, why did we not use here a different generalisation? 123 00:10:17,670 --> 00:10:18,330 Why did we not? 124 00:10:18,330 --> 00:10:18,630 Right. 125 00:10:18,630 --> 00:10:27,960 R is equal to three dimensional R times e r if we want to do this, we have to consider spherical coordinates. 126 00:10:28,710 --> 00:10:36,210 And this we can do if our problem is really sarika symmetric or rotational symmetric in three dimensional 127 00:10:36,210 --> 00:10:36,600 space. 128 00:10:37,470 --> 00:10:44,730 An example in this case would be if you have an electron or some other charge and you put it in the 129 00:10:44,730 --> 00:10:51,570 center of your coordinate system, then the potential or the electric field, they would only depend 130 00:10:51,570 --> 00:10:52,770 on the distance. 131 00:10:53,160 --> 00:10:58,500 So it wouldn't matter at which angle you are looking, it would just matter for the distance. 132 00:10:58,980 --> 00:11:06,020 So in this case, we would have some rotational symmetry and we should use spherical coordinates. 133 00:11:06,930 --> 00:11:14,820 So here we really have the, um, component are, which in this case really is the length of the three 134 00:11:14,820 --> 00:11:15,760 dimensional vector. 135 00:11:16,140 --> 00:11:19,740 So it is X squared plus Y squared plus the square and the square root. 136 00:11:21,390 --> 00:11:27,510 However, this time we do not have a uniform Z component you can see here, it really did. 137 00:11:27,820 --> 00:11:30,510 It really looks different when we go to different spaces. 138 00:11:30,840 --> 00:11:35,000 For example, here, it's very dense, but if we go here, it looks very much different. 139 00:11:36,060 --> 00:11:40,350 And so this makes it a bit more difficult for the other two components. 140 00:11:41,310 --> 00:11:43,800 OK, we can introduce the poll, our angle once again. 141 00:11:43,800 --> 00:11:45,350 So this is the same as previously. 142 00:11:45,360 --> 00:11:52,740 We take this vector here and calculate the polar angle, which is the angle to this X Z plane. 143 00:11:53,310 --> 00:12:00,150 But then we add a second angle, which is this detail here, which is the angle of the vector with respect 144 00:12:00,150 --> 00:12:01,100 to the Z axis. 145 00:12:01,680 --> 00:12:04,980 And this can be calculated by the arcus cosine function. 146 00:12:06,300 --> 00:12:10,530 So what this means is we now have a very, very simple position vector. 147 00:12:10,530 --> 00:12:17,530 It's just the length of the vector times DENR vector, which in this case looks more difficult. 148 00:12:18,270 --> 00:12:21,580 So it's always some advantage and disadvantage. 149 00:12:21,600 --> 00:12:27,720 So here we have a very simple position vector, but the unit vector looks more difficult because we 150 00:12:27,720 --> 00:12:29,130 have no two angles here. 151 00:12:30,240 --> 00:12:32,760 However, in terms of geometry, it's really easy. 152 00:12:33,180 --> 00:12:40,500 So these this vector here are it's really pointing parallel to this right back to our here. 153 00:12:40,800 --> 00:12:47,070 And then we have the two vectors, Eevi and Edyta, and they point along the perpendicular directions. 154 00:12:47,700 --> 00:12:53,700 So once again, if you calculate the scale of products of these vectors and these vectors and all of 155 00:12:53,700 --> 00:12:57,000 ETR and Eevi, it will give zero. 156 00:12:57,060 --> 00:12:59,940 So this means they are all perpendicular to each other. 157 00:13:01,410 --> 00:13:07,440 So in this case, we can write the position vector in these three components, X, Y, Z, and these 158 00:13:07,440 --> 00:13:09,810 are of course these transformations here. 159 00:13:11,930 --> 00:13:17,410 All right, so now we have learned about four different coordinate systems, but most of the time is 160 00:13:17,420 --> 00:13:23,900 at least for physical problems, we will use either Cartesian coordinates or spherical coordinates, 161 00:13:23,930 --> 00:13:26,150 at least if we are in three dimensional space. 162 00:13:27,200 --> 00:13:33,770 So since these spherical coordinates are very important, let us talk about an example and let us I 163 00:13:33,770 --> 00:13:38,020 want to show you something where these coordinate systems are useful. 164 00:13:38,630 --> 00:13:46,190 So I want to show you how to consider integration in spherical coordinates so we will calculate the 165 00:13:46,190 --> 00:13:49,550 volume and the surface area of the sphere.