1 00:00:00,040 --> 00:00:02,460 The following content is provided under a Creative 2 00:00:02,460 --> 00:00:03,870 Commons license. 3 00:00:03,870 --> 00:00:06,320 Your support will help MIT OpenCourseWare 4 00:00:06,320 --> 00:00:10,560 continue to offer high quality educational resources for free. 5 00:00:10,560 --> 00:00:13,300 To make a donation or view additional materials 6 00:00:13,300 --> 00:00:17,210 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,210 --> 00:00:17,862 at ocw.mit.edu. 8 00:00:32,659 --> 00:00:33,430 Hi. 9 00:00:33,430 --> 00:00:36,550 Our lesson today is concerned with introducing 10 00:00:36,550 --> 00:00:40,210 a new type of integral when we deal with calculus 11 00:00:40,210 --> 00:00:41,490 of several variables. 12 00:00:41,490 --> 00:00:43,170 And this might at first glance seem 13 00:00:43,170 --> 00:00:46,390 to be a rather touchy subject, the idea 14 00:00:46,390 --> 00:00:48,000 being that up until now, we've talked 15 00:00:48,000 --> 00:00:50,190 about multiple integration, yet you'll 16 00:00:50,190 --> 00:00:53,070 notice that the title of this block of material 17 00:00:53,070 --> 00:00:54,860 is just called integration. 18 00:00:54,860 --> 00:00:56,360 And the reason for this is that when 19 00:00:56,360 --> 00:00:58,460 we deal with several variables, there 20 00:00:58,460 --> 00:01:00,887 is more than one type of integral, 21 00:01:00,887 --> 00:01:02,970 and what I would like to do is to motivate today's 22 00:01:02,970 --> 00:01:06,430 lesson in terms of a physical example 23 00:01:06,430 --> 00:01:09,860 and then come back and talk about, in more detail, what 24 00:01:09,860 --> 00:01:13,130 happened, and why there is more than one type of integral 25 00:01:13,130 --> 00:01:16,030 in the calculus of several variables. 26 00:01:16,030 --> 00:01:19,620 I called today's lesson Introduction to Line Integrals, 27 00:01:19,620 --> 00:01:23,250 and to visualize how this topic may be introduced, 28 00:01:23,250 --> 00:01:25,270 consider the following situation. 29 00:01:25,270 --> 00:01:32,330 I have a force F, say, defined throughout the xy-plane here. 30 00:01:32,330 --> 00:01:36,830 I have two fixed points, which I'll call P_0 and P_1, 31 00:01:36,830 --> 00:01:41,230 and I pick a particular curve which joins P_0 to P_1. 32 00:01:41,230 --> 00:01:43,680 And the question that I would like to solve 33 00:01:43,680 --> 00:01:47,840 is what is the work done if I move 34 00:01:47,840 --> 00:01:51,900 a particle from P_0 to P_1, along the curve C, 35 00:01:51,900 --> 00:01:54,270 by means of the force f? 36 00:01:54,270 --> 00:01:56,650 And the reason that this is called a line integral 37 00:01:56,650 --> 00:01:58,470 is that somehow or other, you see, 38 00:01:58,470 --> 00:02:00,810 I am going to be dealing with the line 39 00:02:00,810 --> 00:02:04,660 or the curve C. Let me assume, for the sake of argument, 40 00:02:04,660 --> 00:02:07,640 that I know the equation of C in parametric form. 41 00:02:07,640 --> 00:02:09,970 Let's assume simply that C is given 42 00:02:09,970 --> 00:02:13,590 by x as some function of t and y as some function of t, 43 00:02:13,590 --> 00:02:16,250 and since I'm going from P_0 to P_1, 44 00:02:16,250 --> 00:02:19,080 t is defined on a specific closed interval. 45 00:02:19,080 --> 00:02:22,470 Say t goes from t_0 to t_1. 46 00:02:22,470 --> 00:02:25,100 Or, if I want to write the equation of C in vector 47 00:02:25,100 --> 00:02:27,530 form, remember, in vector form, the i 48 00:02:27,530 --> 00:02:30,840 component is x-- or at least in Cartesian coordinate vector 49 00:02:30,840 --> 00:02:35,920 form, the i component is x, the j component is y. 50 00:02:35,920 --> 00:02:37,980 So I can write the same equation, 51 00:02:37,980 --> 00:02:43,460 that the curve is R of t, where R is x of t i plus y of t j. 52 00:02:43,460 --> 00:02:46,580 As long as I'm using Cartesian coordinates here, 53 00:02:46,580 --> 00:02:49,590 let me assume that my force is also given specifically 54 00:02:49,590 --> 00:02:51,660 in terms of Cartesian coordinates. 55 00:02:51,660 --> 00:02:56,700 It's M*i plus N*j, where M and N are specific functions of x 56 00:02:56,700 --> 00:02:57,390 and y. 57 00:02:57,390 --> 00:03:00,450 And just to review, briefly, the problem, 58 00:03:00,450 --> 00:03:05,010 I want to find the work done in moving from P_0 to P_1 along C 59 00:03:05,010 --> 00:03:07,150 under the influence of F. 60 00:03:07,150 --> 00:03:09,170 Now you see, the thing is we already 61 00:03:09,170 --> 00:03:14,275 know how to solve work problems if the distance takes place 62 00:03:14,275 --> 00:03:17,250 along-- if the displacement takes 63 00:03:17,250 --> 00:03:19,570 place along a straight line. 64 00:03:19,570 --> 00:03:21,330 So what we do is the following. 65 00:03:21,330 --> 00:03:26,010 We take the curve C, and we subdivide it into increments. 66 00:03:26,010 --> 00:03:27,640 We break the thing up. 67 00:03:27,640 --> 00:03:30,960 What we do is we look at the k-th increment 68 00:03:30,960 --> 00:03:34,180 and join the two endpoints of that increment 69 00:03:34,180 --> 00:03:39,320 by a straight line, a vector which we'll call delta R sub k. 70 00:03:39,320 --> 00:03:43,600 We pick a point on the curve in this interval 71 00:03:43,600 --> 00:03:46,500 and compute the F. After all, we know the force 72 00:03:46,500 --> 00:03:50,230 as a function of x and y, but we compute that force 73 00:03:50,230 --> 00:03:52,060 for that fixed point. 74 00:03:52,060 --> 00:03:58,110 We then dot that force with this displacement, and that is what? 75 00:03:58,110 --> 00:04:01,860 That's the work that's done if the particle had moved 76 00:04:01,860 --> 00:04:03,810 along the straight line connecting these two 77 00:04:03,810 --> 00:04:09,070 points under the constant force f of x of k star comma y sub k 78 00:04:09,070 --> 00:04:10,330 star. 79 00:04:10,330 --> 00:04:13,060 And then what we do is, to find the approximation, 80 00:04:13,060 --> 00:04:16,120 we add up all of these forces, all 81 00:04:16,120 --> 00:04:19,537 of these works over the various increments. 82 00:04:19,537 --> 00:04:21,870 Notice that you're doing two things in these increments. 83 00:04:21,870 --> 00:04:25,650 You're first of all picking small enough segments 84 00:04:25,650 --> 00:04:28,970 so that you can assume that the straight line approximation is 85 00:04:28,970 --> 00:04:30,890 a good approximation to the curve. 86 00:04:30,890 --> 00:04:33,210 And secondly, small enough segments 87 00:04:33,210 --> 00:04:36,620 so that you're assuming that the force at a given point 88 00:04:36,620 --> 00:04:38,790 is a good approximation to the force 89 00:04:38,790 --> 00:04:40,730 along the entire increment. 90 00:04:40,730 --> 00:04:43,260 At any rate, what we then do is we 91 00:04:43,260 --> 00:04:46,970 sum these up over all the increments. 92 00:04:46,970 --> 00:04:49,390 And by the way, notice that, even though f 93 00:04:49,390 --> 00:04:52,740 is a function of x and y, along the curve C, 94 00:04:52,740 --> 00:04:55,230 x and y are functions of t. 95 00:04:55,230 --> 00:04:59,100 And to emphasize this, let me rewrite this simply 96 00:04:59,100 --> 00:05:03,980 by dividing and multiplying by delta t sub k 97 00:05:03,980 --> 00:05:06,770 to indicate that if I now put this remaining 98 00:05:06,770 --> 00:05:11,270 part of the expression in braces, what's in here 99 00:05:11,270 --> 00:05:15,070 is really some function of t alone. 100 00:05:15,070 --> 00:05:17,740 In other words, notice that this comes 101 00:05:17,740 --> 00:05:19,800 very close to being how we define 102 00:05:19,800 --> 00:05:23,220 the ordinary definite integral when we want to add things up. 103 00:05:23,220 --> 00:05:27,680 You see, t is a scalar, and that the braced expression here 104 00:05:27,680 --> 00:05:29,900 is a function of t alone. 105 00:05:29,900 --> 00:05:32,100 It's a dot product obtained this way. 106 00:05:32,100 --> 00:05:34,800 At any rate, what we're saying now, 107 00:05:34,800 --> 00:05:36,970 quite simply from the physical point of view, 108 00:05:36,970 --> 00:05:38,670 is define the actual work. 109 00:05:38,670 --> 00:05:43,730 Why don't we define the work to simply be the limit of w sub n 110 00:05:43,730 --> 00:05:47,430 as the maximum delta R sub k approaches zero, which 111 00:05:47,430 --> 00:05:48,920 is equivalent to saying what? 112 00:05:48,920 --> 00:05:53,930 Find this limit as delta t sub k, 113 00:05:53,930 --> 00:05:57,410 the maximum delta t sub k approaches zero. 114 00:05:57,410 --> 00:06:00,870 And the point is that this was the ordinary definite integral. 115 00:06:00,870 --> 00:06:03,130 Remember how we wrote this. 116 00:06:03,130 --> 00:06:07,510 You replace the sum by the integral sign. 117 00:06:07,510 --> 00:06:10,940 We assume that t goes from some value t_0 to t_1, 118 00:06:10,940 --> 00:06:13,570 those become our limits of integration. 119 00:06:13,570 --> 00:06:16,520 Every place we see the t sub k star, 120 00:06:16,520 --> 00:06:19,370 we replace it simply by t, and every place 121 00:06:19,370 --> 00:06:22,710 we see the deltas we replace those by d's and we leave off 122 00:06:22,710 --> 00:06:23,780 the subscripts. 123 00:06:23,780 --> 00:06:27,200 So, and I can't emphasize this part strongly enough, 124 00:06:27,200 --> 00:06:31,320 in going from here to here, we are using nothing more 125 00:06:31,320 --> 00:06:33,800 than what happened in part one of our course. 126 00:06:33,800 --> 00:06:37,660 Namely, this is the definition of the definite integral 127 00:06:37,660 --> 00:06:40,820 that we arrive at this expression here. 128 00:06:40,820 --> 00:06:43,710 And to show you other ways in which this is written, 129 00:06:43,710 --> 00:06:48,160 notice that F is M*i plus N*j. 130 00:06:48,160 --> 00:06:54,800 R is x*i plus y*j, so dR/dt is dx/dt i plus dy/dt j. 131 00:06:54,800 --> 00:06:57,860 And so, if we then mechanically just took the dot product here, 132 00:06:57,860 --> 00:07:02,390 notice that the integrand would be M*dx/dt plus N*dy/dt times 133 00:07:02,390 --> 00:07:03,690 dt. 134 00:07:03,690 --> 00:07:08,290 The conventional shortcut for writing this is to cancel dt 135 00:07:08,290 --> 00:07:12,040 from numerator and denominator, and this becomes the standard 136 00:07:12,040 --> 00:07:17,920 notation for the line integral M*dx plus N*dy along the curve 137 00:07:17,920 --> 00:07:20,299 C, where C joins P_0 to P_1. 138 00:07:20,299 --> 00:07:21,840 And as I'll mention in a few minutes, 139 00:07:21,840 --> 00:07:24,690 it's crucial that we specify both the curve 140 00:07:24,690 --> 00:07:26,570 and the points being joined. 141 00:07:26,570 --> 00:07:29,140 And the physicist usually abbreviates this 142 00:07:29,140 --> 00:07:32,110 by coming back to the original notation here, 143 00:07:32,110 --> 00:07:37,000 and when he takes the limit, he just writes this as F dot dR. 144 00:07:37,000 --> 00:07:41,200 And what I'm saying is that this, this, and this are simply 145 00:07:41,200 --> 00:07:45,060 different ways of writing this expression here. 146 00:07:45,060 --> 00:07:45,910 OK. 147 00:07:45,910 --> 00:07:49,180 So this is how this is then evaluated, 148 00:07:49,180 --> 00:07:51,790 and what I'm going to do very shortly 149 00:07:51,790 --> 00:07:57,442 is specifically solve problems using concrete values for F 150 00:07:57,442 --> 00:08:00,010 and P_0 and P_1, but for the time being, 151 00:08:00,010 --> 00:08:01,960 I would like to make a little note here, 152 00:08:01,960 --> 00:08:03,840 to have you see what's happening, 153 00:08:03,840 --> 00:08:05,350 and that's the following. 154 00:08:05,350 --> 00:08:08,426 Notice that double integrals are concerned with regions. 155 00:08:08,426 --> 00:08:10,050 In other words, we use double integrals 156 00:08:10,050 --> 00:08:12,520 to find area, mass, et cetera. 157 00:08:12,520 --> 00:08:16,410 Line integrals are concerned with curves or boundaries 158 00:08:16,410 --> 00:08:16,960 of regions. 159 00:08:16,960 --> 00:08:19,407 In other words, paths that connect two points. 160 00:08:19,407 --> 00:08:20,990 See, in other words, somehow or other, 161 00:08:20,990 --> 00:08:23,490 the double integral is associated with area, 162 00:08:23,490 --> 00:08:25,310 the line integral with length. 163 00:08:25,310 --> 00:08:28,260 And the point is that in one-dimensional space, 164 00:08:28,260 --> 00:08:29,760 the two concepts coincide. 165 00:08:29,760 --> 00:08:32,929 In fact, we don't even talk about area in one dimension, 166 00:08:32,929 --> 00:08:36,770 unless by area we meant one-dimensional area or length. 167 00:08:36,770 --> 00:08:39,400 In other words, both arclength and area 168 00:08:39,400 --> 00:08:41,840 in the typical definition sense of area, 169 00:08:41,840 --> 00:08:43,499 the one-dimensional area, coincide 170 00:08:43,499 --> 00:08:45,280 in one-dimensional space. 171 00:08:45,280 --> 00:08:47,760 In two-dimensional space, they are quite different. 172 00:08:47,760 --> 00:08:50,310 In fact, in a nutshell, to see what's happening, 173 00:08:50,310 --> 00:08:52,930 essentially double integrals are concerned 174 00:08:52,930 --> 00:08:56,980 with what's going on inside a region R, 175 00:08:56,980 --> 00:09:00,930 whereas the line integral somehow or other is concerned 176 00:09:00,930 --> 00:09:03,450 with what's happening as you move along 177 00:09:03,450 --> 00:09:06,020 the boundary of the region R, and we'll 178 00:09:06,020 --> 00:09:09,480 talk about that in more detail later in this lecture, 179 00:09:09,480 --> 00:09:11,990 and this will also be the topic of discussion 180 00:09:11,990 --> 00:09:13,390 in our next lecture. 181 00:09:13,390 --> 00:09:16,310 But essentially, for the remainder of this lesson, 182 00:09:16,310 --> 00:09:19,420 let's take a specific application of our earlier 183 00:09:19,420 --> 00:09:20,940 physical remarks. 184 00:09:20,940 --> 00:09:24,300 In the following examples, let's fix the points P_0 185 00:09:24,300 --> 00:09:29,090 and P_1 to be the points (0, 0) and (1, 1), respectively. 186 00:09:29,090 --> 00:09:35,500 And let the force M*i plus N*j be specifically y squared i 187 00:09:35,500 --> 00:09:37,160 plus x squared j. 188 00:09:37,160 --> 00:09:40,770 In other words, to compute the force at any point x comma 189 00:09:40,770 --> 00:09:44,380 y in the plane, the i component of the force 190 00:09:44,380 --> 00:09:47,530 is just the square of the y-coordinate, 191 00:09:47,530 --> 00:09:49,230 the j component is just the square 192 00:09:49,230 --> 00:09:52,390 of the x-coordinate of the point at which we're 193 00:09:52,390 --> 00:09:53,960 computing the force. 194 00:09:53,960 --> 00:09:56,490 Now, what we're going to do in the next three examples 195 00:09:56,490 --> 00:10:03,300 is simply compute the work done along three different curves 196 00:10:03,300 --> 00:10:08,780 that joined the point P_0 to P_1 and the work done 197 00:10:08,780 --> 00:10:11,920 as a particle moves from P_0 to P_1 198 00:10:11,920 --> 00:10:15,900 under the influence of this specific force, and as I say, 199 00:10:15,900 --> 00:10:19,270 just the curve that joined P_0 and P_1 will be different. 200 00:10:19,270 --> 00:10:22,050 In the first example, the curve C_1 201 00:10:22,050 --> 00:10:25,750 is simply the straight line that joins (0, 0) to (1, 1). 202 00:10:25,750 --> 00:10:28,140 In parametric form, that would be 203 00:10:28,140 --> 00:10:33,090 x equals y equals t where t goes from 0 to 1-- noticing 204 00:10:33,090 --> 00:10:35,750 that when t is 0, you see, x and y are both 0 205 00:10:35,750 --> 00:10:38,720 so we're at the point (0, 0), and when t is 1, 206 00:10:38,720 --> 00:10:42,197 x and y are both 1, so we're at the point (1, 1). 207 00:10:42,197 --> 00:10:44,030 And if we want to write that in vector form, 208 00:10:44,030 --> 00:10:48,430 remembering that the i component is just x and the j component 209 00:10:48,430 --> 00:10:53,170 y, since x and y are both t, the curve has the form r equals t*i 210 00:10:53,170 --> 00:10:55,030 plus t*j. 211 00:10:55,030 --> 00:10:57,820 The work done, writing it symbolically 212 00:10:57,820 --> 00:11:02,050 as we would, in the physical notation, integral along c_1 F 213 00:11:02,050 --> 00:11:04,690 dot dR means what? 214 00:11:04,690 --> 00:11:09,850 We dot F with dR/dt, and multiply that by dt 215 00:11:09,850 --> 00:11:11,520 and integrate that from 0 to 1. 216 00:11:11,520 --> 00:11:13,740 That's what that infinite sum means. 217 00:11:13,740 --> 00:11:16,300 Notice, in this particular case, F, 218 00:11:16,300 --> 00:11:18,860 which is y squared plus x squared, 219 00:11:18,860 --> 00:11:21,205 since x and y are both equal to t, 220 00:11:21,205 --> 00:11:24,920 F is just t squared i plus t squared j. 221 00:11:24,920 --> 00:11:30,430 Since R is t*i plus t*j, dR/dt is just i plus j, 222 00:11:30,430 --> 00:11:34,170 and consequently, the infinite sum that we want to evaluate is 223 00:11:34,170 --> 00:11:39,405 just the integral from 0 to 1-- see, 224 00:11:39,405 --> 00:11:40,780 just taking the dot product here; 225 00:11:40,780 --> 00:11:43,180 t squared plus t squared is 2t squared-- 226 00:11:43,180 --> 00:11:46,350 integral from zero to one, 2 t squared dt, 227 00:11:46,350 --> 00:11:48,885 and using the fundamental theorem of integral calculus 228 00:11:48,885 --> 00:11:53,740 for a single variable, this is simply evaluated by this being 229 00:11:53,740 --> 00:11:57,800 t cubed from 0 to 1-- 2/3 t cubed from 0 to 1, 230 00:11:57,800 --> 00:11:59,730 this is just 2/3. 231 00:11:59,730 --> 00:12:02,230 In other words, you see what's hidden here 232 00:12:02,230 --> 00:12:04,950 is the fact that, when we write the definite integral-- 233 00:12:04,950 --> 00:12:07,280 see, this was a function of two variables in here, 234 00:12:07,280 --> 00:12:10,590 but along the curve, they have functions just of the parameter 235 00:12:10,590 --> 00:12:11,280 t. 236 00:12:11,280 --> 00:12:13,410 This is an ordinary integrand, which 237 00:12:13,410 --> 00:12:16,120 means we can either think of it as an infinite sum 238 00:12:16,120 --> 00:12:20,210 or the antiderivative evaluated in a specific way. 239 00:12:20,210 --> 00:12:22,670 Don't lose track of the fact that the work is still 240 00:12:22,670 --> 00:12:25,560 being defined as a limit, but we can compute it 241 00:12:25,560 --> 00:12:28,660 quite conveniently in terms of the fundamental theorem 242 00:12:28,660 --> 00:12:29,830 of integral calculus. 243 00:12:29,830 --> 00:12:31,880 At any rate, the work done, then, 244 00:12:31,880 --> 00:12:37,020 in example one as we move from P_0 to P_1, under the force F, 245 00:12:37,020 --> 00:12:40,640 along this curve C_1 is just 2/3. 246 00:12:40,640 --> 00:12:44,480 Now what we're going to do is redo the same problem, only 247 00:12:44,480 --> 00:12:47,870 now we are going to choose a different path which connects 248 00:12:47,870 --> 00:12:49,740 (0, 0) to (1, 1). 249 00:12:49,740 --> 00:12:53,240 In fact, I will choose the parabola y equals x squared, 250 00:12:53,240 --> 00:12:55,390 and I'll write it parametrically this way. 251 00:12:55,390 --> 00:12:59,470 I'll let x equal t, y be t squared. 252 00:12:59,470 --> 00:13:01,540 So that says y equals x squared. 253 00:13:01,540 --> 00:13:04,480 And to get to point (0, 0), t must be 1. 254 00:13:04,480 --> 00:13:06,780 To get to point (1, 1), t must be 1. 255 00:13:06,780 --> 00:13:10,360 So again, t varies from 0 to 1 continuously, 256 00:13:10,360 --> 00:13:14,660 or again in vector form, the equation of the curve C_2 is 257 00:13:14,660 --> 00:13:18,250 just what? t*i plus t squared j. 258 00:13:18,250 --> 00:13:21,240 At any rate, the work done, by definition, symbolically, 259 00:13:21,240 --> 00:13:25,060 is the line integral along C_2 F dot dR-- 260 00:13:25,060 --> 00:13:32,420 that still means f dot dR/dt dt-- as t goes from 0 to 1. 261 00:13:32,420 --> 00:13:36,710 In this case, x is t and y is t squared. 262 00:13:36,710 --> 00:13:39,600 So if we come back to our definition of F 263 00:13:39,600 --> 00:13:43,490 and replace x by t and y by t squared, 264 00:13:43,490 --> 00:13:49,630 we see that our force is t to the 1/4 i plus t squared j. 265 00:13:49,630 --> 00:13:55,930 Since R is t*i plus t squared j, dR/dt is just i plus 2t*j. 266 00:13:55,930 --> 00:14:00,110 Remember, we differentiate vectors component by component. 267 00:14:00,110 --> 00:14:04,170 Therefore, the integrand is simply 268 00:14:04,170 --> 00:14:06,520 this dot product, which turns out to be, what? 269 00:14:06,520 --> 00:14:10,590 t to the fourth times 1 plus t squared times 2t, 270 00:14:10,590 --> 00:14:14,600 or t to the fourth plus 2 t cubed. 271 00:14:14,600 --> 00:14:17,150 We now, again, use the fundamental theorem, 272 00:14:17,150 --> 00:14:20,420 evaluate this integral by the antiderivative. 273 00:14:20,420 --> 00:14:23,420 In other words, this is 1/5 t to the fifth plus 1/2 t 274 00:14:23,420 --> 00:14:26,200 to the fourth, evaluated between 0 and 1, 275 00:14:26,200 --> 00:14:28,930 and that answer is 7/10. 276 00:14:28,930 --> 00:14:30,640 In fact, this, I guess, in a way, 277 00:14:30,640 --> 00:14:33,950 was a bad problem from a physical point of view. 278 00:14:33,950 --> 00:14:37,020 An excellent problem from a mathematical point of view. 279 00:14:37,020 --> 00:14:39,840 You see, certainly, from a physical point of view, 280 00:14:39,840 --> 00:14:43,420 7/10 looks enough like 2/3, so the change isn't that drastic. 281 00:14:43,420 --> 00:14:45,850 You can almost call that an experimental error. 282 00:14:45,850 --> 00:14:48,240 But since we weren't doing an experiment here 283 00:14:48,240 --> 00:14:50,120 and we're working with exact numbers, 284 00:14:50,120 --> 00:14:52,050 the fact does remain that, what? 285 00:14:52,050 --> 00:14:56,540 2/3 and 7/10 are different, and if nothing else, what we've 286 00:14:56,540 --> 00:14:57,740 proven is what? 287 00:14:57,740 --> 00:15:02,520 That the work done as we move from two fixed points-- 288 00:15:02,520 --> 00:15:06,310 from the fixed point P_0 to the fixed point P_1, 289 00:15:06,310 --> 00:15:09,370 under the influence of a given force F, 290 00:15:09,370 --> 00:15:13,600 actually does depend on the path that we follow. 291 00:15:13,600 --> 00:15:15,550 You see, we used two different paths, 292 00:15:15,550 --> 00:15:17,270 and got two different answers. 293 00:15:17,270 --> 00:15:22,140 In other words, the line integral M*dx plus N*dy along 294 00:15:22,140 --> 00:15:25,950 the curve C, which joins two fixed points P_0 and P_1, 295 00:15:25,950 --> 00:15:30,080 depends on C as well as on P_0 and P_1, 296 00:15:30,080 --> 00:15:32,331 and that's an important point to keep track of. 297 00:15:32,331 --> 00:15:34,080 We're going to mention this in more detail 298 00:15:34,080 --> 00:15:36,870 as the course goes along. 299 00:15:36,870 --> 00:15:38,590 At any rate though, let that point 300 00:15:38,590 --> 00:15:41,380 remain fixed in abeyance for a while, 301 00:15:41,380 --> 00:15:43,500 and we'll go on with our next example. 302 00:15:43,500 --> 00:15:46,000 And for the next example, what I'm going to do 303 00:15:46,000 --> 00:15:49,660 is pick the same path as an example two, 304 00:15:49,660 --> 00:15:53,510 only in a different parametric form. 305 00:15:53,510 --> 00:15:56,480 Another way of writing the parabola y equals x squared 306 00:15:56,480 --> 00:16:00,200 is to let x equal t squared and y be t the fourth. 307 00:16:00,200 --> 00:16:03,330 You see, in that case, y is still equal to x squared, 308 00:16:03,330 --> 00:16:08,230 and to connect (0, 0) to (1, 1), t still goes from 0 to 1. 309 00:16:08,230 --> 00:16:11,610 So the work done along the curve C_3 310 00:16:11,610 --> 00:16:15,010 is given by this particular symbol. 311 00:16:15,010 --> 00:16:18,560 To evaluate this, remember now, x is t squared. 312 00:16:18,560 --> 00:16:20,970 So x squared will be t to the fourth. 313 00:16:20,970 --> 00:16:23,790 y is t to the fourth, so y squared 314 00:16:23,790 --> 00:16:25,240 would be t to the eighth. 315 00:16:25,240 --> 00:16:27,460 Remembering that our force is y squared i 316 00:16:27,460 --> 00:16:31,000 plus x squared j, written in two-tuple notation, 317 00:16:31,000 --> 00:16:35,280 our force is t to the eighth comma t to the fourth. 318 00:16:35,280 --> 00:16:39,600 Our dR/dt vector is what? 319 00:16:39,600 --> 00:16:43,270 It has components 2t and 4, t cubed, 320 00:16:43,270 --> 00:16:45,970 and we're going to multiply that by dt 321 00:16:45,970 --> 00:16:48,080 and integrate that from 0 to 1. 322 00:16:48,080 --> 00:16:52,670 That's simply 2 t to the ninth plus 4 t to the seventh dt, 323 00:16:52,670 --> 00:16:53,990 from 0 to 1. 324 00:16:53,990 --> 00:16:58,550 That's t to the 10 over 5 plus t to the eighth over 2, 325 00:16:58,550 --> 00:17:01,100 evaluated between 0 and 1, and that's 326 00:17:01,100 --> 00:17:06,230 equal to 7/10, which agrees with this result. 327 00:17:06,230 --> 00:17:07,670 By the way, that's not a proof. 328 00:17:07,670 --> 00:17:10,569 All I've shown is that two parametric forms 329 00:17:10,569 --> 00:17:12,319 led to the same amount of work. 330 00:17:12,319 --> 00:17:14,700 But at any rate, it does seem to indicate 331 00:17:14,700 --> 00:17:18,710 that when you can compute the work along the curve C, 332 00:17:18,710 --> 00:17:22,859 that that work done seems to be independent of the equation 333 00:17:22,859 --> 00:17:23,770 for C. 334 00:17:23,770 --> 00:17:25,560 And hopefully this should be the case, 335 00:17:25,560 --> 00:17:29,630 because we would be in terrible trouble, 336 00:17:29,630 --> 00:17:32,000 I guess is the best word to put this as, 337 00:17:32,000 --> 00:17:35,360 if it turned out that the work done in moving along 338 00:17:35,360 --> 00:17:39,210 a specific curve joining two points under a specific force 339 00:17:39,210 --> 00:17:42,820 depended on the equation by which you represented 340 00:17:42,820 --> 00:17:43,790 the curve. 341 00:17:43,790 --> 00:17:45,780 But I simply wanted to give you an example 342 00:17:45,780 --> 00:17:48,940 to show at least that the mathematical definition appears 343 00:17:48,940 --> 00:17:50,440 to be consistent. 344 00:17:50,440 --> 00:17:53,100 In other words, along the curve C_1, 345 00:17:53,100 --> 00:17:55,790 we got a different amount of work done then 346 00:17:55,790 --> 00:17:58,680 along the curves C_2 or C_3. 347 00:17:58,680 --> 00:18:02,020 But C_2 and C_3 were actually what? 348 00:18:02,020 --> 00:18:05,670 The same curve given by two different equations. 349 00:18:05,670 --> 00:18:08,850 So at any rate, the work does seem to depend on the curve, 350 00:18:08,850 --> 00:18:14,010 but it seems to be independent of the equation by which 351 00:18:14,010 --> 00:18:16,660 the curve is represented. 352 00:18:16,660 --> 00:18:20,430 I think we're in position now to start an overall summary, 353 00:18:20,430 --> 00:18:23,610 going right back to the very crux of the problem as to why 354 00:18:23,610 --> 00:18:26,580 line integrals and multiple integrals exist. 355 00:18:26,580 --> 00:18:29,270 Namely, in particular, in the case 356 00:18:29,270 --> 00:18:33,910 of one independent variable, the integral from a to b f of x dx 357 00:18:33,910 --> 00:18:36,750 has two equivalent interpretations. 358 00:18:36,750 --> 00:18:39,620 And by equivalent, I mean numerically equivalent. 359 00:18:39,620 --> 00:18:41,500 You get the same answer. 360 00:18:41,500 --> 00:18:44,000 They are not equivalent philosophically. 361 00:18:44,000 --> 00:18:45,530 You know, this is one of the reasons 362 00:18:45,530 --> 00:18:47,950 I have this hangup about trying to spell out 363 00:18:47,950 --> 00:18:49,040 the philosophy for you. 364 00:18:49,040 --> 00:18:51,030 In many cases, you will say, we're 365 00:18:51,030 --> 00:18:54,570 interested more in the application than in the theory. 366 00:18:54,570 --> 00:18:56,070 As I may have said to you before, 367 00:18:56,070 --> 00:18:58,420 in fact on many different occasions, 368 00:18:58,420 --> 00:19:01,750 the easy part of mathematics in most cases 369 00:19:01,750 --> 00:19:05,030 is actually applying the computational recipes. 370 00:19:05,030 --> 00:19:09,170 The validity of the recipes is what is usually 371 00:19:09,170 --> 00:19:11,490 the difficult thing, and the biggest mistakes 372 00:19:11,490 --> 00:19:14,890 are not made by a person using the right formula 373 00:19:14,890 --> 00:19:17,010 and making an arithmetic mistake. 374 00:19:17,010 --> 00:19:18,750 It's that he uses the wrong formula 375 00:19:18,750 --> 00:19:21,380 and doesn't make an arithmetic mistake. 376 00:19:21,380 --> 00:19:23,140 So I want this to be very clear to you, 377 00:19:23,140 --> 00:19:26,590 why a certain problem exists in two variables that 378 00:19:26,590 --> 00:19:27,972 didn't exist in one. 379 00:19:27,972 --> 00:19:28,930 And the point is, what? 380 00:19:28,930 --> 00:19:31,510 That conceptually, there were two different ways 381 00:19:31,510 --> 00:19:34,000 to interpret the meaning of the symbol integral 382 00:19:34,000 --> 00:19:36,230 from a to b f of x dx. 383 00:19:36,230 --> 00:19:38,310 Numerically, they came out to be the same, 384 00:19:38,310 --> 00:19:40,560 but what were the two different interpretations? 385 00:19:40,560 --> 00:19:46,660 One was as the limit of a particular infinite sum. 386 00:19:46,660 --> 00:19:49,950 That was the definite integral notation. 387 00:19:49,950 --> 00:19:53,790 The other was, we could visualize the interval from a 388 00:19:53,790 --> 00:20:00,110 to b as being a curve that could be written parametrically, say, 389 00:20:00,110 --> 00:20:05,380 as x is some function of t where t goes from t_0 to t_1, 390 00:20:05,380 --> 00:20:07,920 and we then make the usual change of variables. 391 00:20:07,920 --> 00:20:10,140 The same change of variables that we were talking 392 00:20:10,140 --> 00:20:11,800 about in the previous lecture. 393 00:20:11,800 --> 00:20:17,260 Namely, every place we see x, we replace it by x of t, 394 00:20:17,260 --> 00:20:19,820 and then the correction factor is dx/dt, 395 00:20:19,820 --> 00:20:22,394 and the new limits are from t_0 to t_1. 396 00:20:22,394 --> 00:20:23,560 In other words, we get what? 397 00:20:23,560 --> 00:20:25,600 That the integral from a to b f of x dx 398 00:20:25,600 --> 00:20:32,180 is the integral from t_0 to t_1 f of x of t dx/dt times dt. 399 00:20:32,180 --> 00:20:36,450 Again, observing that this is a bona fide function of t 400 00:20:36,450 --> 00:20:39,910 alone here, so this is just a change of variable type thing. 401 00:20:39,910 --> 00:20:41,070 And what am I saying? 402 00:20:41,070 --> 00:20:42,870 That in the one-dimensional case, 403 00:20:42,870 --> 00:20:46,470 the case of one independent variable, these two concepts, 404 00:20:46,470 --> 00:20:50,460 while different, give rise to the same numerical answer. 405 00:20:50,460 --> 00:20:55,070 In two-space, however, notice that our first interpretation 406 00:20:55,070 --> 00:20:59,950 as an infinite sum translates into the double or multiple 407 00:20:59,950 --> 00:21:00,450 integral. 408 00:21:00,450 --> 00:21:02,010 I keep saying double because we're dealing 409 00:21:02,010 --> 00:21:03,480 with two independent variables. 410 00:21:03,480 --> 00:21:06,070 Obviously by now, I hope it's clear to you 411 00:21:06,070 --> 00:21:08,820 that the results generalize to n variables. 412 00:21:08,820 --> 00:21:10,240 But the idea is, what? 413 00:21:10,240 --> 00:21:14,740 That the first interpretation of the integral in terms 414 00:21:14,740 --> 00:21:17,210 of two independent variables gives 415 00:21:17,210 --> 00:21:21,830 rise to the multiple integral. 416 00:21:21,830 --> 00:21:24,300 In fact, I think I have the-- Yeah, that's right. 417 00:21:24,300 --> 00:21:30,430 And the second interpretation gives rise to this expression 418 00:21:30,430 --> 00:21:31,020 here. 419 00:21:31,020 --> 00:21:35,570 Noticing again that if you have a one-dimensional vector, 420 00:21:35,570 --> 00:21:38,370 the dot product is the ordinary product. 421 00:21:38,370 --> 00:21:41,310 In other words, f of x dx is f of x dot 422 00:21:41,310 --> 00:21:45,240 dx, if you want to think of x as being a one-dimensional vector. 423 00:21:45,240 --> 00:21:46,940 But why is this interpretation? 424 00:21:46,940 --> 00:21:49,060 Notice the change of variables again, 425 00:21:49,060 --> 00:21:54,430 that we think of the curve C as being given by the equation R 426 00:21:54,430 --> 00:21:58,510 equals some function of t as t goes from t_0 to t_1. 427 00:21:58,510 --> 00:21:59,710 So the substitution is what? 428 00:21:59,710 --> 00:22:02,670 This means the particular definite integral, 429 00:22:02,670 --> 00:22:09,200 in terms of t, as t goes from t_0 to t_1, f of R of t 430 00:22:09,200 --> 00:22:13,000 dotted with dR/dt, that again is what? 431 00:22:13,000 --> 00:22:17,220 That again is some scalar function of t, you see? 432 00:22:17,220 --> 00:22:19,577 Times dt. 433 00:22:19,577 --> 00:22:21,410 That's what that second interpretation says, 434 00:22:21,410 --> 00:22:23,060 and that's of course what we mean 435 00:22:23,060 --> 00:22:26,580 by this particular notation. 436 00:22:26,580 --> 00:22:28,340 Two entirely different things. 437 00:22:28,340 --> 00:22:30,460 But at any rate now, getting back to what we saw 438 00:22:30,460 --> 00:22:36,450 in our examples, the line integral C f of R dot dR often 439 00:22:36,450 --> 00:22:40,840 depends on x, but not on the equation which represents C. 440 00:22:40,840 --> 00:22:43,480 By the way, we're going to show a very interesting thing 441 00:22:43,480 --> 00:22:47,390 in the next lecture, as to when the line integral will depend 442 00:22:47,390 --> 00:22:49,760 on C and when it won't. 443 00:22:49,760 --> 00:22:51,400 But the next lecture, actually, is 444 00:22:51,400 --> 00:22:53,730 more general than just answering that question. 445 00:22:53,730 --> 00:22:57,160 The next lecture centers around this question. 446 00:22:57,160 --> 00:22:59,570 Is there any interesting relationship 447 00:22:59,570 --> 00:23:03,690 between line integrals and double integrals? 448 00:23:03,690 --> 00:23:07,740 You see, in the case of one independent variable, 449 00:23:07,740 --> 00:23:09,970 there was a very interesting relationship 450 00:23:09,970 --> 00:23:13,860 between line integrals and so-called double integrals 451 00:23:13,860 --> 00:23:16,300 or integrals of infinite sums. 452 00:23:16,300 --> 00:23:18,442 Namely, they were exactly the same thing. 453 00:23:18,442 --> 00:23:19,650 You couldn't tell them apart. 454 00:23:19,650 --> 00:23:21,480 That's what we meant in our summary 455 00:23:21,480 --> 00:23:24,050 when we said we had two different interpretations 456 00:23:24,050 --> 00:23:27,750 of the same numerical result. Now 457 00:23:27,750 --> 00:23:29,890 the question is, in two independent variables, 458 00:23:29,890 --> 00:23:32,230 these two concepts are different. 459 00:23:32,230 --> 00:23:35,900 Are they related, and if so, how? 460 00:23:35,900 --> 00:23:40,500 And that will be the body of our next lecture. 461 00:23:40,500 --> 00:23:43,130 At any rate then, until next time, goodbye. 462 00:23:46,850 --> 00:23:49,220 Funding for the publication of this video 463 00:23:49,220 --> 00:23:54,100 was provided by the Gabriella and Paul Rosenbaum Foundation. 464 00:23:54,100 --> 00:23:58,270 Help OCW continue to provide free and open access to MIT 465 00:23:58,270 --> 00:24:02,690 courses by making a donation at ocw.mit.edu/donate.