WEBVTT 00:00:00.000 --> 00:00:09.958 align:middle line:90% 00:00:09.958 --> 00:00:11.750 align:middle line:84% PROFESSOR: This is a topic that's sometimes 00:00:11.750 --> 00:00:13.580 align:middle line:90% called calculus of variations. 00:00:13.580 --> 00:00:16.250 align:middle line:90% That's the classic name for it. 00:00:16.250 --> 00:00:20.830 align:middle line:90% 00:00:20.830 --> 00:00:23.785 align:middle line:84% And-- Cookie, are you causing trouble? 00:00:23.785 --> 00:00:26.740 align:middle line:90% 00:00:26.740 --> 00:00:28.710 align:middle line:90% OK. 00:00:28.710 --> 00:00:31.320 align:middle line:84% Calculus of variations, but all it really 00:00:31.320 --> 00:00:37.440 align:middle line:84% is the same idea of derivatives where-- 00:00:37.440 --> 00:00:45.890 align:middle line:84% so we're going to have f of u, a plus du, minus f of u. 00:00:45.890 --> 00:00:48.210 align:middle line:90% That's our df. 00:00:48.210 --> 00:00:49.680 align:middle line:90% Is going to be-- 00:00:49.680 --> 00:00:55.160 align:middle line:84% I'm going to call it f prime of u, du. 00:00:55.160 --> 00:01:06.940 align:middle line:84% It's the same formula, except that now where u is 00:01:06.940 --> 00:01:13.600 align:middle line:90% in a vector space of functions. 00:01:13.600 --> 00:01:17.350 align:middle line:90% 00:01:17.350 --> 00:01:19.780 align:middle line:90% And usually f and-- 00:01:19.780 --> 00:01:23.590 align:middle line:90% and usually f is a scalar. 00:01:23.590 --> 00:01:27.470 align:middle line:90% F of u is just a real number. 00:01:27.470 --> 00:01:30.160 align:middle line:84% So we're to take a function, map it to a real number. 00:01:30.160 --> 00:01:32.440 align:middle line:84% And this is one of the things when I teach 1806, 00:01:32.440 --> 00:01:35.805 align:middle line:84% I come back to, periodically, that vectors are not just 00:01:35.805 --> 00:01:36.430 align:middle line:90% column vectors. 00:01:36.430 --> 00:01:37.480 align:middle line:90% They're even matrices. 00:01:37.480 --> 00:01:40.398 align:middle line:84% You can even think of functions as a vector. 00:01:40.398 --> 00:01:42.190 align:middle line:84% Because you can take two functions, and you 00:01:42.190 --> 00:01:42.910 align:middle line:90% can add them. 00:01:42.910 --> 00:01:44.530 align:middle line:84% You can take two-- and get another function. 00:01:44.530 --> 00:01:46.540 align:middle line:84% You can take two functions and multiply them-- a function 00:01:46.540 --> 00:01:48.500 align:middle line:84% and multiply by 2 and get another function. 00:01:48.500 --> 00:01:49.917 align:middle line:84% And so a lot of the things you can 00:01:49.917 --> 00:01:51.910 align:middle line:84% do in linear algebra for column vectors, 00:01:51.910 --> 00:01:53.080 align:middle line:90% you can do for functions. 00:01:53.080 --> 00:01:55.300 align:middle line:84% You can talk about linear operators on functions. 00:01:55.300 --> 00:01:57.425 align:middle line:84% Like a derivative is a linear operator on function. 00:01:57.425 --> 00:01:59.740 align:middle line:84% You can talk about orthogonal functions, 00:01:59.740 --> 00:02:03.940 align:middle line:84% and dot products of functions, and Gram-Schmidt on functions, 00:02:03.940 --> 00:02:05.813 align:middle line:90% and eigenfunctions. 00:02:05.813 --> 00:02:07.480 align:middle line:84% If you do quantum mechanics, you do lots 00:02:07.480 --> 00:02:09.160 align:middle line:90% of things with eigenfunctions. 00:02:09.160 --> 00:02:11.500 align:middle line:90% So it's the same-- 00:02:11.500 --> 00:02:13.720 align:middle line:90% all the same kinds of concepts. 00:02:13.720 --> 00:02:16.000 align:middle line:84% You have to be a little bit careful with functions. 00:02:16.000 --> 00:02:18.190 align:middle line:84% They can do crazy things that column vectors can't. 00:02:18.190 --> 00:02:19.970 align:middle line:84% Like they can oscillate infinitely fast, 00:02:19.970 --> 00:02:22.100 align:middle line:84% or they can blow up, or do things like that. 00:02:22.100 --> 00:02:24.740 align:middle line:84% So if you want to do rigorous math with functions, 00:02:24.740 --> 00:02:29.180 align:middle line:84% you spend a lot of time nailing down the space of functions 00:02:29.180 --> 00:02:32.870 align:middle line:90% that are not terribly behaved. 00:02:32.870 --> 00:02:34.290 align:middle line:90% I'm going to do that. 00:02:34.290 --> 00:02:36.620 align:middle line:84% I'm just going to be implicitly-- 00:02:36.620 --> 00:02:39.290 align:middle line:84% I'm going to write down an integral of functions-- 00:02:39.290 --> 00:02:40.070 align:middle line:90% of a function. 00:02:40.070 --> 00:02:41.930 align:middle line:84% I'm going to implicitly assume that we're 00:02:41.930 --> 00:02:44.180 align:middle line:84% dealing with functions where that integral is defined. 00:02:44.180 --> 00:02:46.760 align:middle line:90% So things like that. 00:02:46.760 --> 00:02:50.490 align:middle line:90% So let me just do an example. 00:02:50.490 --> 00:03:04.850 align:middle line:84% All right, so let's do u of x is a function from 0 to 1. 00:03:04.850 --> 00:03:06.380 align:middle line:90% X is in 0 to 1. 00:03:06.380 --> 00:03:10.220 align:middle line:90% 00:03:10.220 --> 00:03:12.260 align:middle line:90% Two real numbers. 00:03:12.260 --> 00:03:18.200 align:middle line:84% It's just here's x equals 0, x equals 1, 00:03:18.200 --> 00:03:23.110 align:middle line:90% and u is some function, u of x. 00:03:23.110 --> 00:03:29.770 align:middle line:90% 00:03:29.770 --> 00:03:34.520 align:middle line:90% And let's suppose that f of u-- 00:03:34.520 --> 00:03:38.350 align:middle line:84% I need to write down something that takes in a function 00:03:38.350 --> 00:03:40.490 align:middle line:90% and gives me a number. 00:03:40.490 --> 00:03:42.050 align:middle line:84% And there are a few ways to do that. 00:03:42.050 --> 00:03:50.020 align:middle line:84% But the one that I want to deal with is just integral, 0 to 1, 00:03:50.020 --> 00:03:54.900 align:middle line:90% of say, sine of u of x, dx. 00:03:54.900 --> 00:03:56.220 align:middle line:90% Let's just start with. 00:03:56.220 --> 00:03:58.620 align:middle line:84% That takes in a function, gives me a number. 00:03:58.620 --> 00:04:00.660 align:middle line:84% Technically, it doesn't work for all functions. 00:04:00.660 --> 00:04:05.472 align:middle line:84% You could have functions where this integral doesn't exist 00:04:05.472 --> 00:04:06.430 align:middle line:90% or something like that. 00:04:06.430 --> 00:04:08.940 align:middle line:84% So implicitly, I'm limiting myself 00:04:08.940 --> 00:04:13.000 align:middle line:84% to the space of functions where the integrals work. 00:04:13.000 --> 00:04:18.870 align:middle line:84% And what I want to do is find out what is the-- 00:04:18.870 --> 00:04:22.800 align:middle line:90% what is the gradient of f. 00:04:22.800 --> 00:04:29.490 align:middle line:84% And what is f primed also with respect to u, of course. 00:04:29.490 --> 00:04:31.720 align:middle line:84% So now with respect to a function. 00:04:31.720 --> 00:04:34.170 align:middle line:84% And it's going to be exactly the same definition for it 00:04:34.170 --> 00:04:35.190 align:middle line:90% to take a derivative. 00:04:35.190 --> 00:04:38.100 align:middle line:84% To take a derivative, I just need to linearize. 00:04:38.100 --> 00:04:40.950 align:middle line:84% I need to ask how does f change for a small change 00:04:40.950 --> 00:04:43.822 align:middle line:90% in the input, linearize it. 00:04:43.822 --> 00:04:46.030 align:middle line:84% And then to get-- one step further, to get a gradient 00:04:46.030 --> 00:04:48.240 align:middle line:90% I'm going to need a dot product. 00:04:48.240 --> 00:04:52.770 align:middle line:84% Before we do the gradient and the dot product, 00:04:52.770 --> 00:04:53.940 align:middle line:90% let's just linearize. 00:04:53.940 --> 00:05:03.520 align:middle line:90% So we're going to take df. 00:05:03.520 --> 00:05:05.200 align:middle line:90% We take f of u. 00:05:05.200 --> 00:05:07.530 align:middle line:90% Remember u is a function. 00:05:07.530 --> 00:05:10.560 align:middle line:84% By the way, this is sometimes called a functional. 00:05:10.560 --> 00:05:14.460 align:middle line:90% 00:05:14.460 --> 00:05:16.742 align:middle line:84% It's something that takes in a function 00:05:16.742 --> 00:05:17.700 align:middle line:90% and gives you a number. 00:05:17.700 --> 00:05:20.230 align:middle line:90% 00:05:20.230 --> 00:05:21.210 align:middle line:90% And I'm going to-- 00:05:21.210 --> 00:05:23.130 align:middle line:90% and here I need to-- 00:05:23.130 --> 00:05:28.530 align:middle line:90% f of u plus du, minus f of u. 00:05:28.530 --> 00:05:37.300 align:middle line:84% And the key thing to remember is that this is a function u of x. 00:05:37.300 --> 00:05:38.500 align:middle line:90% That's a terrible x. 00:05:38.500 --> 00:05:46.380 align:middle line:84% And this du here, this is also a function. 00:05:46.380 --> 00:05:54.070 align:middle line:84% You can think of it as a small function, du of x. 00:05:54.070 --> 00:05:58.600 align:middle line:84% And as I said last lecture, to get some notion of small, 00:05:58.600 --> 00:06:00.550 align:middle line:84% we're going to need a norm of a function. 00:06:00.550 --> 00:06:03.050 align:middle line:84% But pretty soon, I'm going to have dot products of functions 00:06:03.050 --> 00:06:03.250 align:middle line:90% So. 00:06:03.250 --> 00:06:04.310 align:middle line:90% I'll have norms as well. 00:06:04.310 --> 00:06:06.160 align:middle line:84% So let me sweep that under the rug. 00:06:06.160 --> 00:06:08.320 align:middle line:84% But it's just like a little perturbation. 00:06:08.320 --> 00:06:11.350 align:middle line:90% So you're taking u of x-- 00:06:11.350 --> 00:06:14.010 align:middle line:90% 00:06:14.010 --> 00:06:20.400 align:middle line:84% u of x, and you're adding some arbitrary little change 00:06:20.400 --> 00:06:20.940 align:middle line:90% to this. 00:06:20.940 --> 00:06:25.700 align:middle line:90% This is a u plus du of x. 00:06:25.700 --> 00:06:28.840 align:middle line:84% So you're going to change that function by a little bit 00:06:28.840 --> 00:06:31.350 align:middle line:90% anywhere you want. 00:06:31.350 --> 00:06:32.620 align:middle line:90% It has to be small. 00:06:32.620 --> 00:06:35.625 align:middle line:84% And so we can drop terms that are like du squared. 00:06:35.625 --> 00:06:39.420 align:middle line:90% 00:06:39.420 --> 00:06:41.070 align:middle line:90% And so what is this so this? 00:06:41.070 --> 00:06:43.915 align:middle line:90% This is integral from 0 to 1. 00:06:43.915 --> 00:06:46.310 align:middle line:90% And I said f was sine of ud-- 00:06:46.310 --> 00:06:47.670 align:middle line:90% of xdx. 00:06:47.670 --> 00:06:56.340 align:middle line:84% So this is looks like sine of u of x, plus du 00:06:56.340 --> 00:07:03.900 align:middle line:90% of x, minus sine of x, dx. 00:07:03.900 --> 00:07:05.940 align:middle line:84% And I'm just plugging that into my f. 00:07:05.940 --> 00:07:08.920 align:middle line:90% My f was integral sign u. 00:07:08.920 --> 00:07:11.470 align:middle line:84% And I'm going to ask, how much did that integral change 00:07:11.470 --> 00:07:15.320 align:middle line:84% for a small change in u, and I want to linearize it. 00:07:15.320 --> 00:07:17.260 align:middle line:90% So I want to write this as-- 00:07:17.260 --> 00:07:19.735 align:middle line:84% drop every term that goes like du squared or higher. 00:07:19.735 --> 00:07:22.555 align:middle line:90% 00:07:22.555 --> 00:07:25.570 align:middle line:84% And, of course, the point is all only need to do 00:07:25.570 --> 00:07:28.120 align:middle line:84% is linearize this, linearize the integrand. 00:07:28.120 --> 00:07:33.190 align:middle line:90% 00:07:33.190 --> 00:07:37.630 align:middle line:84% That thing, sine-- sorry-- and this is sine of x. 00:07:37.630 --> 00:07:38.770 align:middle line:90% This is sine of u of x. 00:07:38.770 --> 00:07:43.537 align:middle line:90% 00:07:43.537 --> 00:07:45.860 align:middle line:90% OK, so I want to linearize this. 00:07:45.860 --> 00:07:46.740 align:middle line:90% What is sine? 00:07:46.740 --> 00:07:54.820 align:middle line:84% So sine-- sine of u plus du, that's 00:07:54.820 --> 00:08:00.880 align:middle line:84% equal to sine of u plus the derivative, which 00:08:00.880 --> 00:08:04.135 align:middle line:90% is cosine of u times du. 00:08:04.135 --> 00:08:08.970 align:middle line:90% 00:08:08.970 --> 00:08:15.960 align:middle line:84% This is just-- du is a function, but du of x is just a number. 00:08:15.960 --> 00:08:18.570 align:middle line:84% At each x, I have a small change in u of x. 00:08:18.570 --> 00:08:26.000 align:middle line:84% So this is just ordinary 18.01, linearization, 00:08:26.000 --> 00:08:30.020 align:middle line:84% or from lecture one of this class. 00:08:30.020 --> 00:08:31.108 align:middle line:90% Or this could be-- 00:08:31.108 --> 00:08:31.650 align:middle line:90% yeah, anyway. 00:08:31.650 --> 00:08:33.692 align:middle line:84% So this is just a number, ordinary linearization. 00:08:33.692 --> 00:08:39.850 align:middle line:84% And then, of course, this term here cancels that term. 00:08:39.850 --> 00:08:45.080 align:middle line:84% So what this is is integral 0 to 1 00:08:45.080 --> 00:08:53.810 align:middle line:84% of what's left-- cosine of u of x, du, dx. 00:08:53.810 --> 00:08:56.867 align:middle line:90% And this is exactly-- 00:08:56.867 --> 00:08:57.950 align:middle line:90% this is a linear operator. 00:08:57.950 --> 00:09:07.840 align:middle line:84% This is f primed of u adding on du. 00:09:07.840 --> 00:09:12.380 align:middle line:84% Sorry, and again, I should put a du of x here. 00:09:12.380 --> 00:09:13.000 align:middle line:90% Is that clear? 00:09:13.000 --> 00:09:19.357 align:middle line:84% If I plug in any function here, any change in my u, this 00:09:19.357 --> 00:09:20.440 align:middle line:90% is clearly linear in that. 00:09:20.440 --> 00:09:22.930 align:middle line:84% If I double this, it doubles the integral. 00:09:22.930 --> 00:09:24.430 align:middle line:84% If I add two things here, it adds 00:09:24.430 --> 00:09:25.610 align:middle line:90% two things in the integral. 00:09:25.610 --> 00:09:27.455 align:middle line:90% So this is a linear operator. 00:09:27.455 --> 00:09:27.955 align:middle line:90% We're done. 00:09:27.955 --> 00:09:34.060 align:middle line:90% 00:09:34.060 --> 00:09:40.090 align:middle line:84% That takes du in, and it gives you out a number. 00:09:40.090 --> 00:09:43.350 align:middle line:90% this is my df. 00:09:43.350 --> 00:09:47.850 align:middle line:84% So once you get used to this as being 00:09:47.850 --> 00:09:52.470 align:middle line:84% the definition of a derivative, you 00:09:52.470 --> 00:09:54.120 align:middle line:90% can always fall back on this. 00:09:54.120 --> 00:09:57.660 align:middle line:84% When u is some crazy object, matrix, function, whatever, 00:09:57.660 --> 00:10:00.540 align:middle line:84% as long as it's a vector space, or technically a Banach space-- 00:10:00.540 --> 00:10:03.540 align:middle line:90% it has to have a norm-- 00:10:03.540 --> 00:10:05.238 align:middle line:84% then you can define the derivative 00:10:05.238 --> 00:10:06.780 align:middle line:84% as the linear operator that gives you 00:10:06.780 --> 00:10:07.738 align:middle line:90% the first-order change. 00:10:07.738 --> 00:10:10.480 align:middle line:90% 00:10:10.480 --> 00:10:14.460 align:middle line:84% So, as I said, technically we're dropping 00:10:14.460 --> 00:10:19.540 align:middle line:84% terms of little o, du, terms that go to 0 faster than du 00:10:19.540 --> 00:10:20.040 align:middle line:90% here. 00:10:20.040 --> 00:10:23.550 align:middle line:84% So to make that precise, you need to have a norm. 00:10:23.550 --> 00:10:28.448 align:middle line:84% But anyway, this is a function that takes a vector-- 00:10:28.448 --> 00:10:30.990 align:middle line:84% by function, I'm thinking of it as a vector space-- a vector, 00:10:30.990 --> 00:10:32.340 align:middle line:90% and it gives you out a scalar. 00:10:32.340 --> 00:10:36.000 align:middle line:84% So I should be able to write that linear operator, 00:10:36.000 --> 00:10:39.000 align:middle line:84% this linear form, as a dot product with something. 00:10:39.000 --> 00:10:41.397 align:middle line:84% So for that, also, I want a dot product, 00:10:41.397 --> 00:10:42.480 align:middle line:90% which will give me a norm. 00:10:42.480 --> 00:10:44.100 align:middle line:90% So that's just to remind you-- 00:10:44.100 --> 00:10:46.950 align:middle line:84% hopefully, you've seen this before in 1806 00:10:46.950 --> 00:10:48.570 align:middle line:90% or some other linear algebra-- 00:10:48.570 --> 00:10:50.240 align:middle line:90% dot products are functions. 00:10:50.240 --> 00:11:00.420 align:middle line:90% 00:11:00.420 --> 00:11:02.925 align:middle line:90% So-- yes? 00:11:02.925 --> 00:11:04.785 align:middle line:84% STUDENT: I have a quick question. 00:11:04.785 --> 00:11:06.570 align:middle line:84% One takeaway that I have so far is 00:11:06.570 --> 00:11:09.073 align:middle line:84% that a derivative is something that 00:11:09.073 --> 00:11:12.117 align:middle line:84% lives in the dual space or the thing, that u. 00:11:12.117 --> 00:11:13.590 align:middle line:90% Could the input be a function? 00:11:13.590 --> 00:11:14.340 align:middle line:90% PROFESSOR: Yeah. 00:11:14.340 --> 00:11:19.260 align:middle line:90% Well, if the output is a scalar. 00:11:19.260 --> 00:11:23.250 align:middle line:84% Yes, so this is exactly something in the dual space. 00:11:23.250 --> 00:11:26.046 align:middle line:84% Physicists would call this a bra. 00:11:26.046 --> 00:11:27.960 align:middle line:84% STUDENT: But then, if this is the function, 00:11:27.960 --> 00:11:30.472 align:middle line:84% I would expect that the derivative be something 00:11:30.472 --> 00:11:31.730 align:middle line:90% like distribution and measure? 00:11:31.730 --> 00:11:34.122 align:middle line:84% Or should I think of this as like the cosine of-- 00:11:34.122 --> 00:11:35.580 align:middle line:84% could it be like the density of it? 00:11:35.580 --> 00:11:37.038 align:middle line:84% PROFESSOR: Yeah, this is a measure. 00:11:37.038 --> 00:11:40.150 align:middle line:84% Because-- exactly, this is the density of that measure. 00:11:40.150 --> 00:11:40.650 align:middle line:90% Yeah. 00:11:40.650 --> 00:11:41.160 align:middle line:90% STUDENT: Thank you. 00:11:41.160 --> 00:11:42.200 align:middle line:90% PROFESSOR: Absolutely. 00:11:42.200 --> 00:11:44.520 align:middle line:84% This takes the measure of that function. 00:11:44.520 --> 00:11:45.510 align:middle line:90% Exactly, yeah. 00:11:45.510 --> 00:11:47.205 align:middle line:90% So there's all sorts of-- 00:11:47.205 --> 00:11:49.530 align:middle line:84% you can think of this as a linear form, 00:11:49.530 --> 00:11:52.290 align:middle line:84% or something in the dual space, or a covector, this vector 00:11:52.290 --> 00:11:54.570 align:middle line:90% space. 00:11:54.570 --> 00:11:56.910 align:middle line:84% But I want to say a little bit informal in terms 00:11:56.910 --> 00:11:58.690 align:middle line:90% of those abstractions here. 00:11:58.690 --> 00:12:01.840 align:middle line:84% And as I said, technically speaking, 00:12:01.840 --> 00:12:04.350 align:middle line:84% we have to restrict ourselves to not all possible functions 00:12:04.350 --> 00:12:04.680 align:middle line:90% here. 00:12:04.680 --> 00:12:07.055 align:middle line:84% They have to be functions where this integral is defined. 00:12:07.055 --> 00:12:10.410 align:middle line:84% So they can't be functions that blow up, or are not measurable, 00:12:10.410 --> 00:12:12.930 align:middle line:90% or something like that. 00:12:12.930 --> 00:12:15.960 align:middle line:84% But that's something that pure mathematicians worry about that 00:12:15.960 --> 00:12:17.280 align:middle line:90% physicists and engineers don't. 00:12:17.280 --> 00:12:20.370 align:middle line:84% Because the functions-- basically, 00:12:20.370 --> 00:12:22.440 align:middle line:84% if this is a physical context, implicitly you're 00:12:22.440 --> 00:12:24.690 align:middle line:84% dealing with functions where these things are defined. 00:12:24.690 --> 00:12:26.940 align:middle line:84% Like you only really worry about ruling out 00:12:26.940 --> 00:12:30.060 align:middle line:84% those crazy functions if you're doing pure math that's 00:12:30.060 --> 00:12:33.090 align:middle line:90% unmoored from reality. 00:12:33.090 --> 00:12:35.640 align:middle line:90% So dot products are functions. 00:12:35.640 --> 00:12:37.830 align:middle line:84% So if you take a dot product with two column 00:12:37.830 --> 00:12:39.333 align:middle line:90% vectors, the standard-- 00:12:39.333 --> 00:12:42.000 align:middle line:84% there are many dot products, but the standard one, the Euclidean 00:12:42.000 --> 00:12:44.860 align:middle line:84% one, is you just multiply the components and add them up. 00:12:44.860 --> 00:12:49.080 align:middle line:84% So if you think of two functions-- 00:12:49.080 --> 00:12:58.570 align:middle line:84% u of x and v of x on x in 0 to 1, 00:12:58.570 --> 00:13:04.170 align:middle line:90% then the Euclidean dot product-- 00:13:04.170 --> 00:13:11.352 align:middle line:84% the obvious dot product, the Euclidean inner product, 00:13:11.352 --> 00:13:13.560 align:middle line:84% should be to multiply the components and add them up. 00:13:13.560 --> 00:13:15.980 align:middle line:84% So the components, if you think of these as column vectors-- 00:13:15.980 --> 00:13:18.022 align:middle line:84% they're like infinite dimensional column vectors. 00:13:18.022 --> 00:13:20.640 align:middle line:84% The components are the values at each point. 00:13:20.640 --> 00:13:25.760 align:middle line:84% So basically, what you want to do is multiply u of x times v 00:13:25.760 --> 00:13:28.730 align:middle line:84% of x and add that up for all the x's. 00:13:28.730 --> 00:13:31.610 align:middle line:84% And the analog of adding it up is an integral. 00:13:31.610 --> 00:13:34.290 align:middle line:90% 00:13:34.290 --> 00:13:40.088 align:middle line:84% That's our u dot v. So we can-- and, of course, this 00:13:40.088 --> 00:13:41.630 align:middle line:84% is not the only possible dot product, 00:13:41.630 --> 00:13:43.630 align:middle line:84% but this is the most obvious one-- the first one 00:13:43.630 --> 00:13:46.420 align:middle line:90% that you typically reach for. 00:13:46.420 --> 00:13:48.530 align:middle line:84% So once you have a dot product, then, of course, 00:13:48.530 --> 00:13:51.200 align:middle line:90% we have a norm. 00:13:51.200 --> 00:13:54.220 align:middle line:84% So if you want the norm of u, that's 00:13:54.220 --> 00:13:57.350 align:middle line:84% just the square root of the integral from 0 00:13:57.350 --> 00:14:02.370 align:middle line:90% to 1 of u of x, square dx. 00:14:02.370 --> 00:14:04.870 align:middle line:84% I guess I don't need absolute values here because everything 00:14:04.870 --> 00:14:05.740 align:middle line:90% is real right now. 00:14:05.740 --> 00:14:08.770 align:middle line:84% If they were complex, I'd be putting complex conjugates 00:14:08.770 --> 00:14:11.680 align:middle line:90% over one of these. 00:14:11.680 --> 00:14:14.800 align:middle line:84% So let me remove my absolute values there, just for-- 00:14:14.800 --> 00:14:16.300 align:middle line:90% I don't need them. 00:14:16.300 --> 00:14:19.210 align:middle line:84% So this is-- and once we have the norm, 00:14:19.210 --> 00:14:22.540 align:middle line:84% then that's the precise sense in which we're dropping 00:14:22.540 --> 00:14:25.270 align:middle line:90% terms that go like du squared. 00:14:25.270 --> 00:14:28.120 align:middle line:84% So we're going to drop terms that 00:14:28.120 --> 00:14:31.450 align:middle line:84% go like the norm of du squared and higher, or actually, 00:14:31.450 --> 00:14:33.580 align:middle line:90% anything higher than linear. 00:14:33.580 --> 00:14:39.880 align:middle line:84% And once you have this, now we can compute-- 00:14:39.880 --> 00:14:41.230 align:middle line:90% we can find the gradient. 00:14:41.230 --> 00:14:47.000 align:middle line:90% 00:14:47.000 --> 00:14:48.470 align:middle line:90% Because what? 00:14:48.470 --> 00:14:49.730 align:middle line:90% Was our df? 00:14:49.730 --> 00:15:03.190 align:middle line:84% Our df, which was f primed of u acting on due, 00:15:03.190 --> 00:15:10.330 align:middle line:84% we said that was integral from 0 to 1 of cosine 00:15:10.330 --> 00:15:18.820 align:middle line:90% of u of x, times du of x, dx. 00:15:18.820 --> 00:15:21.610 align:middle line:84% We want to think of this-- just our generic definition 00:15:21.610 --> 00:15:24.850 align:middle line:84% of the gradient is when you have something that 00:15:24.850 --> 00:15:26.650 align:middle line:84% takes a vector in and scalar out, 00:15:26.650 --> 00:15:31.270 align:middle line:84% the first derivative had better be the dot product 00:15:31.270 --> 00:15:36.980 align:middle line:90% of the gradient with du. 00:15:36.980 --> 00:15:46.050 align:middle line:84% And just by inspection, this is-- 00:15:46.050 --> 00:15:49.870 align:middle line:84% here, this is our definition of a dot product. 00:15:49.870 --> 00:15:51.960 align:middle line:90% So what is the gradient? 00:15:51.960 --> 00:15:57.060 align:middle line:90% 00:15:57.060 --> 00:16:02.145 align:middle line:84% What would I take the dot product with du of to get this? 00:16:02.145 --> 00:16:05.050 align:middle line:90% 00:16:05.050 --> 00:16:08.610 align:middle line:90% Cos of u-- exactly. 00:16:08.610 --> 00:16:09.900 align:middle line:90% So it's a function. 00:16:09.900 --> 00:16:12.690 align:middle line:90% It's just cosine of u of x. 00:16:12.690 --> 00:16:15.750 align:middle line:90% 00:16:15.750 --> 00:16:20.970 align:middle line:84% By the way, in this context, this 00:16:20.970 --> 00:16:22.980 align:middle line:84% is called-- this is sometimes also called 00:16:22.980 --> 00:16:24.360 align:middle line:90% a Frechet derivative. 00:16:24.360 --> 00:16:30.390 align:middle line:90% 00:16:30.390 --> 00:16:32.520 align:middle line:84% Frechet derivative technically refers just 00:16:32.520 --> 00:16:37.238 align:middle line:84% to this general notion of if you have any Banach space, 00:16:37.238 --> 00:16:39.780 align:middle line:84% the Frechet derivative is the linear operator that does this. 00:16:39.780 --> 00:16:43.140 align:middle line:84% But most often, you see this coming up 00:16:43.140 --> 00:16:45.500 align:middle line:84% in the context of vector spaces of functions. 00:16:45.500 --> 00:16:47.250 align:middle line:84% So that's where they use that terminology. 00:16:47.250 --> 00:16:49.590 align:middle line:84% For everything else, people just say the derivative. 00:16:49.590 --> 00:16:55.680 align:middle line:84% But it really is just the derivative, but that's it. 00:16:55.680 --> 00:16:57.040 align:middle line:90% This is a gradient. 00:16:57.040 --> 00:17:00.120 align:middle line:84% So let's do a more complicated case. 00:17:00.120 --> 00:17:05.325 align:middle line:84% So let's do another example, a more tricky example. 00:17:05.325 --> 00:17:11.790 align:middle line:90% 00:17:11.790 --> 00:17:26.579 align:middle line:84% Suppose that-- suppose that f of u-- 00:17:26.579 --> 00:17:30.300 align:middle line:84% so we're still going to be functions from 0 to 1. 00:17:30.300 --> 00:17:39.680 align:middle line:84% Suppose it's the integral from 0 to 1 over the square root of 1 00:17:39.680 --> 00:17:44.930 align:middle line:90% plus u primed of x squared, dx. 00:17:44.930 --> 00:17:53.820 align:middle line:84% Where this is just the derivative, the ordinary du, 00:17:53.820 --> 00:17:58.150 align:middle line:90% dx, the 18.01 derivative. 00:17:58.150 --> 00:18:01.337 align:middle line:84% So this takes a function in, gives you a number out. 00:18:01.337 --> 00:18:03.670 align:middle line:84% Clearly, we have to restrict ourselves to differentiable 00:18:03.670 --> 00:18:04.660 align:middle line:90% functions now. 00:18:04.660 --> 00:18:10.810 align:middle line:90% 00:18:10.810 --> 00:18:12.610 align:middle line:90% And what is this? 00:18:12.610 --> 00:18:14.050 align:middle line:90% Does anyone recognize this? 00:18:14.050 --> 00:18:19.430 align:middle line:84% Can you describe what this quantity is geometrically? 00:18:19.430 --> 00:18:23.930 align:middle line:84% So if I have a function u of x, and it goes-- 00:18:23.930 --> 00:18:28.230 align:middle line:90% 00:18:28.230 --> 00:18:31.590 align:middle line:90% here is u your x-- 00:18:31.590 --> 00:18:35.100 align:middle line:84% what is this quantity that I've just computed here? 00:18:35.100 --> 00:18:37.710 align:middle line:84% From first year calculus, there's 00:18:37.710 --> 00:18:39.000 align:middle line:90% an important name of this. 00:18:39.000 --> 00:18:40.550 align:middle line:90% Yeah? 00:18:40.550 --> 00:18:41.570 align:middle line:90% It's the length. 00:18:41.570 --> 00:18:42.597 align:middle line:90% It's the arc length. 00:18:42.597 --> 00:18:43.805 align:middle line:90% It's the length of the curve. 00:18:43.805 --> 00:18:50.820 align:middle line:90% 00:18:50.820 --> 00:19:00.890 align:middle line:84% The arc length of u of x from x equals 0 to x equals 1. 00:19:00.890 --> 00:19:02.580 align:middle line:90% That's what this quantity is. 00:19:02.580 --> 00:19:05.130 align:middle line:84% If you multiply through by dx, this 00:19:05.130 --> 00:19:09.690 align:middle line:84% is basically square root of dx squared plus dy squared. 00:19:09.690 --> 00:19:11.610 align:middle line:84% If you take-- each little line segment 00:19:11.610 --> 00:19:13.860 align:middle line:84% here it's like the length of that little line segment, 00:19:13.860 --> 00:19:16.260 align:middle line:90% and you're adding them up. 00:19:16.260 --> 00:19:20.900 align:middle line:84% So now let's think about the derivative. 00:19:20.900 --> 00:19:25.990 align:middle line:90% So now let's take f of u. 00:19:25.990 --> 00:19:31.580 align:middle line:84% And we're going to add a little change to u. 00:19:31.580 --> 00:19:33.950 align:middle line:90% Subtract f of u. 00:19:33.950 --> 00:19:36.170 align:middle line:84% So it's always good to just fall back 00:19:36.170 --> 00:19:39.620 align:middle line:84% on the definition, the most basic definition 00:19:39.620 --> 00:19:42.970 align:middle line:90% if we're in doubt of what to do. 00:19:42.970 --> 00:19:46.220 align:middle line:84% And we're going to just add a little change. 00:19:46.220 --> 00:19:50.450 align:middle line:90% And what's this going to be? 00:19:50.450 --> 00:19:53.510 align:middle line:84% So let's just-- it's the integral 00:19:53.510 --> 00:20:02.570 align:middle line:84% of from 0 to 1, the square root of 1, plus u plus du, 00:20:02.570 --> 00:20:06.620 align:middle line:84% primed, squared, minus the square root of 1 00:20:06.620 --> 00:20:09.530 align:middle line:90% plus u primed and squared. 00:20:09.530 --> 00:20:13.180 align:middle line:90% So, again, we just think-- 00:20:13.180 --> 00:20:15.580 align:middle line:84% so now we have to Taylor expand this. 00:20:15.580 --> 00:20:18.460 align:middle line:84% We have to assume that du is now a differentiable thing, 00:20:18.460 --> 00:20:20.900 align:middle line:84% and its derivative is getting smaller and smaller as well. 00:20:20.900 --> 00:20:26.410 align:middle line:84% So I can do this kind of expansion-- first-order 00:20:26.410 --> 00:20:28.540 align:middle line:90% expansion. 00:20:28.540 --> 00:20:29.750 align:middle line:90% So what is this? 00:20:29.750 --> 00:20:31.800 align:middle line:90% Well, it's the-- if I have-- 00:20:31.800 --> 00:20:45.030 align:middle line:84% let's see, if I have 1, plus v plus dv, squared, 00:20:45.030 --> 00:20:47.520 align:middle line:84% what's the Taylor expansion of that? 00:20:47.520 --> 00:20:51.130 align:middle line:84% 1 plus v squared, that's the first term. 00:20:51.130 --> 00:20:52.770 align:middle line:84% And the second term is the derivative 00:20:52.770 --> 00:20:56.700 align:middle line:84% with respect to v, which ends up being v over square root of 1 00:20:56.700 --> 00:20:59.250 align:middle line:90% plus v squared, times dv. 00:20:59.250 --> 00:21:02.640 align:middle line:84% I went through that a little bit. 00:21:02.640 --> 00:21:04.530 align:middle line:90% But this is just 18.01-- 00:21:04.530 --> 00:21:07.980 align:middle line:90% this is just 18.01, so plus. 00:21:07.980 --> 00:21:12.550 align:middle line:84% This is this function for dv equals 0. 00:21:12.550 --> 00:21:14.007 align:middle line:90% This is the derivative of this. 00:21:14.007 --> 00:21:16.090 align:middle line:84% So when we take the derivative of the square root, 00:21:16.090 --> 00:21:19.240 align:middle line:84% it's 1/2, 1 over the square root. 00:21:19.240 --> 00:21:20.990 align:middle line:84% But then there's a derivative of this, 00:21:20.990 --> 00:21:25.900 align:middle line:84% which gives you a 2v on top, so the two cancels the 1/2. 00:21:25.900 --> 00:21:34.300 align:middle line:84% So this just-- so just from calculus and expanding 00:21:34.300 --> 00:21:36.820 align:middle line:84% everything to first order in du and also 00:21:36.820 --> 00:21:38.530 align:middle line:84% du primed-- so dropping anything that 00:21:38.530 --> 00:21:41.080 align:middle line:84% goes like-- anything with a du, whether you 00:21:41.080 --> 00:21:42.580 align:middle line:84% take a derivative or not, squared, 00:21:42.580 --> 00:21:44.020 align:middle line:90% we're going to just drop. 00:21:44.020 --> 00:21:49.190 align:middle line:84% This is the integral of that second term, 00:21:49.190 --> 00:21:59.340 align:middle line:84% which is u over square root of 1 plus u squared, times-- 00:21:59.340 --> 00:22:00.930 align:middle line:90% and du, that's primed. 00:22:00.930 --> 00:22:05.825 align:middle line:90% 00:22:05.825 --> 00:22:12.870 align:middle line:90% and times du primed dx. 00:22:12.870 --> 00:22:17.318 align:middle line:90% 00:22:17.318 --> 00:22:29.020 align:middle line:84% So this is a linear operator, but it's acting on du primed, 00:22:29.020 --> 00:22:33.670 align:middle line:90% not on du, per se. 00:22:33.670 --> 00:22:35.890 align:middle line:84% The derivative is a linear operator on du, 00:22:35.890 --> 00:22:40.120 align:middle line:84% but I'd like to write it directly in terms of du. 00:22:40.120 --> 00:22:42.040 align:middle line:84% Because I'd like to write it eventually as-- 00:22:42.040 --> 00:22:44.515 align:middle line:84% take a gradient, write it as a dot product of something 00:22:44.515 --> 00:22:45.520 align:middle line:90% with du. 00:22:45.520 --> 00:22:49.360 align:middle line:84% So what I really want to do is write this 00:22:49.360 --> 00:22:51.520 align:middle line:90% as a dot product of a gradient-- 00:22:51.520 --> 00:22:54.950 align:middle line:90% 00:22:54.950 --> 00:22:57.380 align:middle line:90% dot du. 00:22:57.380 --> 00:23:03.020 align:middle line:84% And somehow, I have to get a du and not a du primed. 00:23:03.020 --> 00:23:04.940 align:middle line:84% So we just saw this in the first hour. 00:23:04.940 --> 00:23:07.440 align:middle line:84% If you have an integral, and you have 00:23:07.440 --> 00:23:09.440 align:middle line:84% a term that has a derivative, and you don't want 00:23:09.440 --> 00:23:13.680 align:middle line:90% the derivative, what do you do? 00:23:13.680 --> 00:23:15.560 align:middle line:84% What can you do in first year calculus 00:23:15.560 --> 00:23:19.200 align:middle line:84% to get rid of a derivative on a term in an integral? 00:23:19.200 --> 00:23:20.340 align:middle line:90% Integrate by parts. 00:23:20.340 --> 00:23:30.450 align:middle line:90% 00:23:30.450 --> 00:23:33.530 align:middle line:90% So what do we do? 00:23:33.530 --> 00:23:35.695 align:middle line:84% So that whole thing, let's see df. 00:23:35.695 --> 00:23:38.330 align:middle line:90% 00:23:38.330 --> 00:23:40.040 align:middle line:84% There is a boundary term that looks 00:23:40.040 --> 00:23:43.070 align:middle line:84% like u primed over square root of 1, 00:23:43.070 --> 00:23:50.090 align:middle line:84% plus u prime squared times due at the boundaries, 0, 1. 00:23:50.090 --> 00:23:54.530 align:middle line:84% And then you have a minus sign, and an integral 00:23:54.530 --> 00:23:57.790 align:middle line:90% from 0 to 1 of-- 00:23:57.790 --> 00:24:08.560 align:middle line:84% we take this whole thing primed, du, and then times dx. 00:24:08.560 --> 00:24:10.540 align:middle line:84% All the u's and the du's are functions of x. 00:24:10.540 --> 00:24:16.740 align:middle line:90% 00:24:16.740 --> 00:24:20.880 align:middle line:90% So to go further, this is it. 00:24:20.880 --> 00:24:22.980 align:middle line:90% This is the linear operator. 00:24:22.980 --> 00:24:32.135 align:middle line:84% This is now our-- this is our f prime of u acting on du. 00:24:32.135 --> 00:24:33.980 align:middle line:84% And this is a linear operator acting on du. 00:24:33.980 --> 00:24:35.230 align:middle line:90% And there's no du primes here. 00:24:35.230 --> 00:24:36.765 align:middle line:90% This is nice. 00:24:36.765 --> 00:24:38.140 align:middle line:84% But it has these-- these boundary 00:24:38.140 --> 00:24:40.420 align:middle line:90% terms are kind of annoying. 00:24:40.420 --> 00:24:43.315 align:middle line:84% But usually, in the context where I'm interested in this, 00:24:43.315 --> 00:24:45.440 align:middle line:84% I'm actually going to set some boundary conditions. 00:24:45.440 --> 00:24:53.065 align:middle line:84% So in particular, suppose I want to fix the endpoints of my u. 00:24:53.065 --> 00:24:57.020 align:middle line:84% And I want to find the curve that 00:24:57.020 --> 00:24:59.820 align:middle line:84% has the minimum distance from one point to another. 00:24:59.820 --> 00:25:01.910 align:middle line:84% Of course, that's going to be a straight line. 00:25:01.910 --> 00:25:04.370 align:middle line:90% But suppose I didn't know that. 00:25:04.370 --> 00:25:06.800 align:middle line:84% This is a simple problem where you know the solution ahead 00:25:06.800 --> 00:25:08.675 align:middle line:84% of time, but the more complicated things, you 00:25:08.675 --> 00:25:11.590 align:middle line:90% won't know that. 00:25:11.590 --> 00:25:14.110 align:middle line:84% So I want to-- suppose we want to fix-- 00:25:14.110 --> 00:25:19.140 align:middle line:90% 00:25:19.140 --> 00:25:26.040 align:middle line:84% to fix-- let's just even say u of 0 equals u of 1, 00:25:26.040 --> 00:25:28.100 align:middle line:90% equals 0, just for simplicity. 00:25:28.100 --> 00:25:32.280 align:middle line:90% 00:25:32.280 --> 00:25:45.620 align:middle line:84% And now we want to find the u that minimizes the arc length f 00:25:45.620 --> 00:25:48.030 align:middle line:90% of u. 00:25:48.030 --> 00:25:51.780 align:middle line:84% We know it's a straight line from one point to the other, 00:25:51.780 --> 00:25:55.500 align:middle line:84% but let's pretend we don't know that. 00:25:55.500 --> 00:25:57.660 align:middle line:84% So what we want-- if you want to minimize 00:25:57.660 --> 00:26:00.765 align:middle line:84% something, what you want is for the gradient to be 0. 00:26:00.765 --> 00:26:03.330 align:middle line:90% 00:26:03.330 --> 00:26:05.310 align:middle line:84% So we're going to take a gradient set 00:26:05.310 --> 00:26:07.890 align:middle line:90% it equal to 0 in a minute. 00:26:07.890 --> 00:26:14.610 align:middle line:84% But if we fix the endpoints, we're 00:26:14.610 --> 00:26:21.750 align:middle line:84% going to want du also at 0 or 1 to be 0. 00:26:21.750 --> 00:26:26.940 align:middle line:84% If we fix them at 0-- because if we start with our vector 00:26:26.940 --> 00:26:29.110 align:middle line:84% space of u is at our 0 at the endpoints, 00:26:29.110 --> 00:26:31.360 align:middle line:84% and we want to look at not all possible perturbations, 00:26:31.360 --> 00:26:33.310 align:middle line:84% but perturbations that keep those end fixed. 00:26:33.310 --> 00:26:35.310 align:middle line:84% Or if I kept the ends fixed at some other point, 00:26:35.310 --> 00:26:39.690 align:middle line:84% still I would want to look at not all possible perturbations, 00:26:39.690 --> 00:26:41.797 align:middle line:84% I'd want to look at the perturbations that 00:26:41.797 --> 00:26:42.630 align:middle line:90% keep the ends fixed. 00:26:42.630 --> 00:26:45.000 align:middle line:84% So the u is 0 at the ends, so this 00:26:45.000 --> 00:26:48.065 align:middle line:90% keeps the end points fixed. 00:26:48.065 --> 00:26:52.330 align:middle line:90% 00:26:52.330 --> 00:26:55.710 align:middle line:84% And then what happens is that boundary term disappears. 00:26:55.710 --> 00:27:00.700 align:middle line:90% 00:27:00.700 --> 00:27:03.010 align:middle line:84% What was it-- u primed, over square root 00:27:03.010 --> 00:27:09.030 align:middle line:84% of 1 plus u prime squared, du taken at 0, 1. 00:27:09.030 --> 00:27:14.420 align:middle line:84% This is now just 0 because we fixed the endpoints. 00:27:14.420 --> 00:27:24.740 align:middle line:84% So now df is just integral of 0 to 1 of u primed, 00:27:24.740 --> 00:27:27.560 align:middle line:84% over square root of 1 plus u prime squared, 00:27:27.560 --> 00:27:31.400 align:middle line:90% primed, times du, dx. 00:27:31.400 --> 00:27:41.720 align:middle line:84% So our gradient is just this thing here. 00:27:41.720 --> 00:27:43.970 align:middle line:84% It's the thing we're taking with the dot product of du 00:27:43.970 --> 00:27:45.840 align:middle line:90% with to get the solution. 00:27:45.840 --> 00:27:49.040 align:middle line:84% So it's just u primed, over square root of 1 00:27:49.040 --> 00:27:54.580 align:middle line:90% plus u primed squared, primed. 00:27:54.580 --> 00:27:55.630 align:middle line:90% Does everyone see that? 00:27:55.630 --> 00:27:58.190 align:middle line:90% 00:27:58.190 --> 00:28:01.970 align:middle line:84% So this df had better be the dot product of grad f 00:28:01.970 --> 00:28:04.683 align:middle line:90% with dot grad u. 00:28:04.683 --> 00:28:05.600 align:middle line:90% I integrated by parts. 00:28:05.600 --> 00:28:07.498 align:middle line:84% So I turned my u primed into a du. 00:28:07.498 --> 00:28:10.040 align:middle line:84% I set boundary conditions to fix the boundary so the boundary 00:28:10.040 --> 00:28:10.940 align:middle line:90% terms disappeared. 00:28:10.940 --> 00:28:16.130 align:middle line:84% What I have left is just a dot product of this with the u. 00:28:16.130 --> 00:28:18.920 align:middle line:84% So I can think of this as the gradient. 00:28:18.920 --> 00:28:23.930 align:middle line:84% And if you're at a minimum, it's the same-- for the same reason 00:28:23.930 --> 00:28:27.260 align:middle line:84% as in 18.01, and 18.02, and so forth, at a minimum, 00:28:27.260 --> 00:28:28.670 align:middle line:90% the gradient has to be 0. 00:28:28.670 --> 00:28:31.768 align:middle line:84% Because if it's not 0, you can move in that direction 00:28:31.768 --> 00:28:34.310 align:middle line:84% and make it bigger, or move in the opposite of that direction 00:28:34.310 --> 00:28:35.143 align:middle line:90% and make it smaller. 00:28:35.143 --> 00:28:37.580 align:middle line:90% 00:28:37.580 --> 00:28:41.900 align:middle line:84% So this has to equal 0 at a minimum. 00:28:41.900 --> 00:28:45.880 align:middle line:90% 00:28:45.880 --> 00:28:48.010 align:middle line:90% And so now let's just-- 00:28:48.010 --> 00:28:48.940 align:middle line:90% I need to work out-- 00:28:48.940 --> 00:28:52.060 align:middle line:84% OK, now, it's a little bit of a calculus exercise. 00:28:52.060 --> 00:28:56.320 align:middle line:84% What is the derivative of this thing? 00:28:56.320 --> 00:28:57.760 align:middle line:90% OK, so now, it's quotient rule. 00:28:57.760 --> 00:29:03.190 align:middle line:84% So this is going to be quotient rule. 00:29:03.190 --> 00:29:05.600 align:middle line:84% So on the quotient rule, on the bottom, 00:29:05.600 --> 00:29:07.160 align:middle line:84% you get that denominator squared. 00:29:07.160 --> 00:29:09.550 align:middle line:84% So you get 1 plus u prime squared, 00:29:09.550 --> 00:29:11.290 align:middle line:90% the square root squared. 00:29:11.290 --> 00:29:13.270 align:middle line:84% On the top, you get the derivative 00:29:13.270 --> 00:29:16.090 align:middle line:84% of the top, which is u double primed 00:29:16.090 --> 00:29:21.660 align:middle line:84% times the bottom, square root of 1 plus u primed squared. 00:29:21.660 --> 00:29:25.380 align:middle line:90% 00:29:25.380 --> 00:29:33.210 align:middle line:84% And then minus the top times the derivative of the bottom. 00:29:33.210 --> 00:29:34.320 align:middle line:90% And we just did that. 00:29:34.320 --> 00:29:36.487 align:middle line:84% The derivative of the derivative of that square root 00:29:36.487 --> 00:29:39.360 align:middle line:84% is, in fact, u primed, over square root of 1 00:29:39.360 --> 00:29:42.030 align:middle line:90% plus u primed squared, I think. 00:29:42.030 --> 00:29:45.530 align:middle line:90% 00:29:45.530 --> 00:29:48.230 align:middle line:84% Hopefully, I'm not making any mistakes here. 00:29:48.230 --> 00:29:51.680 align:middle line:84% OK, so now, if I want to put this over-- 00:29:51.680 --> 00:29:53.760 align:middle line:84% let's simplify this a little bit. 00:29:53.760 --> 00:29:57.270 align:middle line:90% This is 1 plus u prime squared. 00:29:57.270 --> 00:29:58.268 align:middle line:90% I'm just going to-- 00:29:58.268 --> 00:29:59.810 align:middle line:84% I'm going to put a 3/2 on the bottom. 00:29:59.810 --> 00:30:02.850 align:middle line:84% So I'm going to pull this term and put it on the bottom. 00:30:02.850 --> 00:30:07.050 align:middle line:84% So I'm just going to multiply and divide by that square root. 00:30:07.050 --> 00:30:14.210 align:middle line:84% And what you get is a u primed, times 1 plus u prime squared, 00:30:14.210 --> 00:30:16.400 align:middle line:90% minus u prime squared. 00:30:16.400 --> 00:30:19.760 align:middle line:90% 00:30:19.760 --> 00:30:22.805 align:middle line:84% I'm just simplifying this a little bit. 00:30:22.805 --> 00:30:25.180 align:middle line:84% STUDENT: That could be used on the second one. 00:30:25.180 --> 00:30:26.560 align:middle line:90% PROFESSOR: The second one? 00:30:26.560 --> 00:30:30.440 align:middle line:84% No, Because this came from u primed times u primed. 00:30:30.440 --> 00:30:32.063 align:middle line:90% This one is-- 00:30:32.063 --> 00:30:32.969 align:middle line:90% STUDENT: No, no. 00:30:32.969 --> 00:30:35.240 align:middle line:90% The u prime that is-- 00:30:35.240 --> 00:30:36.930 align:middle line:90% PROFESSOR: There? 00:30:36.930 --> 00:30:38.280 align:middle line:90% That came from this square root. 00:30:38.280 --> 00:30:39.480 align:middle line:84% I multiplied the top and the bottom 00:30:39.480 --> 00:30:41.460 align:middle line:84% by a square root of 1 plus u prime squared. 00:30:41.460 --> 00:30:43.420 align:middle line:84% STUDENT: OK, and then the second term. 00:30:43.420 --> 00:30:44.175 align:middle line:90% PROFESSOR: The second term-- 00:30:44.175 --> 00:30:44.430 align:middle line:90% STUDENT: The u prime. 00:30:44.430 --> 00:30:45.570 align:middle line:90% PROFESSOR: What's that? 00:30:45.570 --> 00:30:49.280 align:middle line:84% STUDENT: The u prime actually stays in the form-- 00:30:49.280 --> 00:30:50.530 align:middle line:90% PROFESSOR: So wait, this term? 00:30:50.530 --> 00:30:52.300 align:middle line:84% Are we talking about the right term or the left term? 00:30:52.300 --> 00:30:54.290 align:middle line:84% STUDENT: We're talking about the term on the right. 00:30:54.290 --> 00:30:55.115 align:middle line:90% PROFESSOR: This one here? 00:30:55.115 --> 00:30:56.843 align:middle line:84% STUDENT: Yeah, we're talking about the u 00:30:56.843 --> 00:31:04.208 align:middle line:84% prime squared because if it is 1 plus u prime squared, 00:31:04.208 --> 00:31:07.170 align:middle line:90% you would-- 00:31:07.170 --> 00:31:11.545 align:middle line:84% PROFESSOR: Sorry, are you saying this term is wrong? 00:31:11.545 --> 00:31:12.420 align:middle line:90% Did I make a mistake? 00:31:12.420 --> 00:31:13.660 align:middle line:90% STUDENT: Yeah, I think. 00:31:13.660 --> 00:31:16.050 align:middle line:84% PROFESSOR: So was this term correct? 00:31:16.050 --> 00:31:18.300 align:middle line:84% STUDENT: I don't think it is u prime. 00:31:18.300 --> 00:31:19.230 align:middle line:90% PROFESSOR: Oh, yes. 00:31:19.230 --> 00:31:19.830 align:middle line:90% Thank you. 00:31:19.830 --> 00:31:20.550 align:middle line:90% Oh good. 00:31:20.550 --> 00:31:22.710 align:middle line:84% Good, because that's going to screw things up. 00:31:22.710 --> 00:31:28.130 align:middle line:90% OK, so-- yes, OK. 00:31:28.130 --> 00:31:29.750 align:middle line:90% So good, sorry. 00:31:29.750 --> 00:31:30.650 align:middle line:90% Yes, you're right. 00:31:30.650 --> 00:31:32.000 align:middle line:90% The top is not even that. 00:31:32.000 --> 00:31:33.710 align:middle line:90% It's u double primed-- 00:31:33.710 --> 00:31:34.328 align:middle line:90% good. 00:31:34.328 --> 00:31:36.620 align:middle line:84% OK, I knew something was supposed to cancel here, good. 00:31:36.620 --> 00:31:39.890 align:middle line:84% It's actually u double primed times u primed. 00:31:39.890 --> 00:31:40.920 align:middle line:90% Right? 00:31:40.920 --> 00:31:41.420 align:middle line:90% OK, good. 00:31:41.420 --> 00:31:44.000 align:middle line:84% Because it's the derivative of the u primed squared. 00:31:44.000 --> 00:31:46.490 align:middle line:84% It gives you a 2 u double primed, u primed. 00:31:46.490 --> 00:31:48.860 align:middle line:84% The two cancel the derivative of the square root. 00:31:48.860 --> 00:31:49.940 align:middle line:90% Good. 00:31:49.940 --> 00:31:52.850 align:middle line:84% Good, I'm missing a u double primed there. 00:31:52.850 --> 00:31:54.560 align:middle line:84% Good, I knew it was supposed to simplify. 00:31:54.560 --> 00:31:55.977 align:middle line:84% I was a little worried about that. 00:31:55.977 --> 00:31:59.650 align:middle line:84% So now this term cancels this term. 00:31:59.650 --> 00:32:03.370 align:middle line:84% And what you get is just u double primed, 00:32:03.370 --> 00:32:07.030 align:middle line:84% over 1 plus u primed squared to the 3/2. 00:32:07.030 --> 00:32:11.910 align:middle line:84% And this is supposed to equal 0, we said. 00:32:11.910 --> 00:32:16.740 align:middle line:84% So the only way it can 0, is if u double primed equals 0. 00:32:16.740 --> 00:32:19.620 align:middle line:90% 00:32:19.620 --> 00:32:21.000 align:middle line:90% And that's a straight line. 00:32:21.000 --> 00:32:27.820 align:middle line:90% 00:32:27.820 --> 00:32:30.190 align:middle line:84% So what we just derived is that the shortest distance 00:32:30.190 --> 00:32:31.840 align:middle line:84% between two points is a straight line. 00:32:31.840 --> 00:32:34.810 align:middle line:90% Of course, we knew that. 00:32:34.810 --> 00:32:37.360 align:middle line:90% But let's do a generalization. 00:32:37.360 --> 00:32:42.690 align:middle line:90% 00:32:42.690 --> 00:32:52.020 align:middle line:84% OK, so let's say f of u is an integral from a 00:32:52.020 --> 00:32:55.500 align:middle line:90% to b of some other function. 00:32:55.500 --> 00:32:59.550 align:middle line:90% Let's call it capital F of-- 00:32:59.550 --> 00:33:01.350 align:middle line:90% it may depend on u. 00:33:01.350 --> 00:33:03.390 align:middle line:90% It may depend on u primed. 00:33:03.390 --> 00:33:05.250 align:middle line:90% It may depend on x. 00:33:05.250 --> 00:33:08.950 align:middle line:90% 00:33:08.950 --> 00:33:11.470 align:middle line:90% This is a functional. 00:33:11.470 --> 00:33:12.875 align:middle line:90% And then what is df? 00:33:12.875 --> 00:33:16.300 align:middle line:90% 00:33:16.300 --> 00:33:24.760 align:middle line:84% So df is f of u plus du, minus f of u. 00:33:24.760 --> 00:33:28.220 align:middle line:84% So now what are we going to have? 00:33:28.220 --> 00:33:32.090 align:middle line:84% So there is going to be a term that looks 00:33:32.090 --> 00:33:40.370 align:middle line:90% like partial f, partial u-- 00:33:40.370 --> 00:33:42.620 align:middle line:84% this an ordinary 18.01 partial derivative-- 00:33:42.620 --> 00:33:43.790 align:middle line:90% I'm assuming u is a number-- 00:33:43.790 --> 00:33:45.720 align:middle line:84% I guess it could be a vector or something like that, 00:33:45.720 --> 00:33:47.060 align:middle line:90% but let's just keep it simple-- 00:33:47.060 --> 00:33:48.230 align:middle line:90% times du. 00:33:48.230 --> 00:33:52.020 align:middle line:90% 00:33:52.020 --> 00:33:58.980 align:middle line:84% There is also a term that looks like partial f, partial u 00:33:58.980 --> 00:34:06.910 align:middle line:84% primed, times du primed, times dx. 00:34:06.910 --> 00:34:10.480 align:middle line:84% And this sometimes confuses people. 00:34:10.480 --> 00:34:11.695 align:middle line:90% It's like how can I-- 00:34:11.695 --> 00:34:14.380 align:middle line:84% if I take derivatives with a u, keeping u primed fixed, 00:34:14.380 --> 00:34:16.340 align:middle line:84% because obviously, u and u are related. 00:34:16.340 --> 00:34:18.340 align:middle line:84% It really means the derivative of f with respect 00:34:18.340 --> 00:34:20.440 align:middle line:90% to the first argument. 00:34:20.440 --> 00:34:23.560 align:middle line:84% This is the derivative of f respect to the second argument. 00:34:23.560 --> 00:34:25.870 align:middle line:84% And then you plug in u and u primed afterwards. 00:34:25.870 --> 00:34:28.060 align:middle line:84% So it's just-- it's just doing exactly the same. 00:34:28.060 --> 00:34:29.893 align:middle line:84% And you're doing much like this in homework, 00:34:29.893 --> 00:34:32.409 align:middle line:84% but I'm skipping a couple of steps here. 00:34:32.409 --> 00:34:36.880 align:middle line:84% This change-- there's a change that comes from the first term, 00:34:36.880 --> 00:34:39.097 align:middle line:84% like we did on the first example, a sine of u. 00:34:39.097 --> 00:34:41.139 align:middle line:84% There's a change that comes from the second term, 00:34:41.139 --> 00:34:42.670 align:middle line:90% like in the second example. 00:34:42.670 --> 00:34:44.260 align:middle line:90% And you have to add them up. 00:34:44.260 --> 00:34:47.724 align:middle line:90% But now you integrate by parts. 00:34:47.724 --> 00:34:52.860 align:middle line:90% 00:34:52.860 --> 00:34:57.390 align:middle line:84% And on the du primed term, because we 00:34:57.390 --> 00:34:59.700 align:middle line:84% don't like having du primes, and so you 00:34:59.700 --> 00:35:08.781 align:middle line:84% get a partial f, partial u primed, times du at a, b. 00:35:08.781 --> 00:35:13.305 align:middle line:84% And then we have a plus an integral a to b. 00:35:13.305 --> 00:35:17.210 align:middle line:84% And there is a term that looks like df, du. 00:35:17.210 --> 00:35:19.190 align:middle line:84% And there's another term that looks 00:35:19.190 --> 00:35:27.710 align:middle line:84% like, with a minus sign, df, du primed, primed, 00:35:27.710 --> 00:35:31.340 align:middle line:90% multiplying du times dx. 00:35:31.340 --> 00:35:36.950 align:middle line:84% Again, du is-- my small change is a du of x. 00:35:36.950 --> 00:35:38.780 align:middle line:84% I'm just suppressing all of this. 00:35:38.780 --> 00:35:43.280 align:middle line:84% And this term, we're going to set to 0 00:35:43.280 --> 00:35:45.170 align:middle line:90% if the endpoints are fixed-- 00:35:45.170 --> 00:35:49.520 align:middle line:90% 00:35:49.520 --> 00:35:52.390 align:middle line:90% of u are fixed. 00:35:52.390 --> 00:35:54.720 align:middle line:84% So if we keep u fixed at the endpoints, 00:35:54.720 --> 00:35:56.880 align:middle line:84% then we're going to only allow changes, 00:35:56.880 --> 00:35:58.440 align:middle line:84% du's, that are arbitrary functions, 00:35:58.440 --> 00:36:00.360 align:middle line:84% but at the endpoints, they have to be 0. 00:36:00.360 --> 00:36:04.920 align:middle line:84% And so what you get is this term here is the gradient. 00:36:04.920 --> 00:36:14.850 align:middle line:90% 00:36:14.850 --> 00:36:17.130 align:middle line:90% This is a grad f. 00:36:17.130 --> 00:36:18.433 align:middle line:90% This is a dot product with due. 00:36:18.433 --> 00:36:19.600 align:middle line:90% And so I can write this out. 00:36:19.600 --> 00:36:29.870 align:middle line:84% So this is-- so grad f is df, du, minus df, 00:36:29.870 --> 00:36:33.850 align:middle line:90% du primed, primed. 00:36:33.850 --> 00:36:38.410 align:middle line:84% And we set this equal to 0 at an extremum, 00:36:38.410 --> 00:36:41.060 align:middle line:90% at a minimum and maximum. 00:36:41.060 --> 00:36:45.760 align:middle line:90% And the result, this is-- 00:36:45.760 --> 00:36:50.276 align:middle line:84% we set this equal to 0, this is some differential equation-- 00:36:50.276 --> 00:36:53.540 align:middle line:90% 00:36:53.540 --> 00:36:56.860 align:middle line:90% equation in u. 00:36:56.860 --> 00:36:59.660 align:middle line:90% 00:36:59.660 --> 00:37:04.590 align:middle line:84% It involves-- and it's actually a second order. 00:37:04.590 --> 00:37:08.250 align:middle line:84% Because you started out with u and u primed in your f, 00:37:08.250 --> 00:37:10.740 align:middle line:84% but when you take this derivative here, 00:37:10.740 --> 00:37:13.087 align:middle line:84% I'm going to get some u double primed terms. 00:37:13.087 --> 00:37:15.420 align:middle line:84% Does anyone know the name of this differential equation? 00:37:15.420 --> 00:37:18.320 align:middle line:90% Has anyone seen this before? 00:37:18.320 --> 00:37:20.630 align:middle line:84% Euler-Lagrange equation-- so these 00:37:20.630 --> 00:37:23.870 align:middle line:84% are exactly the Euler-Lagrange equations. 00:37:23.870 --> 00:37:27.178 align:middle line:84% Where did you see Euler Lagrange equations before? 00:37:27.178 --> 00:37:28.220 align:middle line:90% STUDENT: Optimal control. 00:37:28.220 --> 00:37:29.345 align:middle line:90% PROFESSOR: Optimal control. 00:37:29.345 --> 00:37:33.380 align:middle line:90% 00:37:33.380 --> 00:37:37.700 align:middle line:84% But you also famously see them in advanced mechanics, 00:37:37.700 --> 00:37:39.710 align:middle line:90% like 806 or-- 00:37:39.710 --> 00:37:42.320 align:middle line:84% I don't know what the number is these days. 00:37:42.320 --> 00:37:44.840 align:middle line:84% Where this is-- there's something 00:37:44.840 --> 00:37:46.580 align:middle line:84% called the least action principle. 00:37:46.580 --> 00:37:49.880 align:middle line:84% And you can write down the equations of motion 00:37:49.880 --> 00:37:51.170 align:middle line:90% as minimizing some integral. 00:37:51.170 --> 00:37:52.940 align:middle line:90% And you get these Euler-- 00:37:52.940 --> 00:37:54.820 align:middle line:84% instead of Newton's laws, you get-- 00:37:54.820 --> 00:37:56.750 align:middle line:84% as a different way of getting Newton's laws, 00:37:56.750 --> 00:37:58.260 align:middle line:84% you get these Euler-Lagrange equations. 00:37:58.260 --> 00:37:59.330 align:middle line:84% They're really nice because they let 00:37:59.330 --> 00:38:01.010 align:middle line:84% you write down basically Newton's 00:38:01.010 --> 00:38:05.720 align:middle line:84% laws in really weird coordinate systems pretty easily. 00:38:05.720 --> 00:38:07.910 align:middle line:84% Yeah, so there's a famous example 00:38:07.910 --> 00:38:11.660 align:middle line:84% of the Euler-Lagrange equation that everyone gives, 00:38:11.660 --> 00:38:13.800 align:middle line:84% which is called the brachistochrone problem. 00:38:13.800 --> 00:38:15.800 align:middle line:84% Has anyone heard of the brachistochrone problem? 00:38:15.800 --> 00:38:17.420 align:middle line:84% We have one person-- a couple people. 00:38:17.420 --> 00:38:19.250 align:middle line:84% So this is a famous problem from, I think, 00:38:19.250 --> 00:38:21.080 align:middle line:90% the 18th century. 00:38:21.080 --> 00:38:26.560 align:middle line:84% So suppose you have a mass falling down a slope, 00:38:26.560 --> 00:38:30.320 align:middle line:84% just sliding down a curve with gravity. 00:38:30.320 --> 00:38:35.270 align:middle line:84% So it's going down this curve, like a ramp with no friction. 00:38:35.270 --> 00:38:36.440 align:middle line:90% It just slides down. 00:38:36.440 --> 00:38:39.240 align:middle line:84% And you're starting here, and you're ending up here. 00:38:39.240 --> 00:38:42.170 align:middle line:84% And the question is, what shape should this ramp 00:38:42.170 --> 00:38:44.930 align:middle line:84% be to get down as quickly as possible? 00:38:44.930 --> 00:38:47.330 align:middle line:90% Should it be a straight ramp? 00:38:47.330 --> 00:38:48.740 align:middle line:90% Or should it be like this? 00:38:48.740 --> 00:38:50.420 align:middle line:84% You can pretty easily see that it shouldn't be like this 00:38:50.420 --> 00:38:52.610 align:middle line:84% because then it'll start out going very, very slowly 00:38:52.610 --> 00:38:53.810 align:middle line:90% and then go-- 00:38:53.810 --> 00:38:57.290 align:middle line:84% a straight ramp, or should it go down very steeply 00:38:57.290 --> 00:38:58.550 align:middle line:90% and then flatten out? 00:38:58.550 --> 00:39:01.310 align:middle line:84% If it goes down very steeply, it'll get going-- 00:39:01.310 --> 00:39:03.470 align:middle line:84% get a good head of speed pretty quickly, but then 00:39:03.470 --> 00:39:04.580 align:middle line:90% it has a longer distance. 00:39:04.580 --> 00:39:06.622 align:middle line:84% The straight line would be the shortest distance, 00:39:06.622 --> 00:39:08.120 align:middle line:90% but it's not the highest speed. 00:39:08.120 --> 00:39:11.180 align:middle line:84% And so it was a famous problem to find-- 00:39:11.180 --> 00:39:12.890 align:middle line:84% called the brachistochrone problem-- 00:39:12.890 --> 00:39:16.790 align:middle line:84% to find the curve of this ramp that minimizes the time. 00:39:16.790 --> 00:39:18.080 align:middle line:90% And it's exactly an integral. 00:39:18.080 --> 00:39:19.100 align:middle line:90% You can write down this. 00:39:19.100 --> 00:39:21.380 align:middle line:84% And you write down a differential equation. 00:39:21.380 --> 00:39:22.970 align:middle line:84% You can solve it to get this curve. 00:39:22.970 --> 00:39:24.970 align:middle line:84% Unfortunately, the differential equation you get 00:39:24.970 --> 00:39:26.120 align:middle line:90% is rather nasty. 00:39:26.120 --> 00:39:30.440 align:middle line:84% It is solvable in closed form, but it's 00:39:30.440 --> 00:39:33.260 align:middle line:84% nothing I would ever assign as a homework problem in 1803, 00:39:33.260 --> 00:39:34.260 align:middle line:90% for example. 00:39:34.260 --> 00:39:36.090 align:middle line:84% So it turns out that the minimum shaped 00:39:36.090 --> 00:39:38.940 align:middle line:90% curve is called a cycloid. 00:39:38.940 --> 00:39:40.590 align:middle line:90% It's the curve that if you-- 00:39:40.590 --> 00:39:44.010 align:middle line:84% if you have a wheel rolling, and you attach a pen 00:39:44.010 --> 00:39:47.100 align:middle line:84% to one spot in the wheel, and as it rolls, 00:39:47.100 --> 00:39:51.510 align:middle line:84% you mark out the curve that wheel traces out, that's 00:39:51.510 --> 00:39:53.520 align:middle line:90% the shape that you get. 00:39:53.520 --> 00:39:54.780 align:middle line:90% So it's a famous-- 00:39:54.780 --> 00:40:00.210 align:middle line:84% so I'll even write that down, to google example-- 00:40:00.210 --> 00:40:03.240 align:middle line:90% 00:40:03.240 --> 00:40:08.170 align:middle line:84% google the brachistochrone problem-- 00:40:08.170 --> 00:40:14.460 align:middle line:90% brachistochrone problem. 00:40:14.460 --> 00:40:16.960 align:middle line:90% That's one example. 00:40:16.960 --> 00:40:25.605 align:middle line:84% Another example would be google the principle of least action. 00:40:25.605 --> 00:40:31.672 align:middle line:90% 00:40:31.672 --> 00:40:33.630 align:middle line:84% That's how you get the Euler-Lagrange equations 00:40:33.630 --> 00:40:35.370 align:middle line:90% in mechanics. 00:40:35.370 --> 00:40:36.720 align:middle line:90% So a great example. 00:40:36.720 --> 00:40:38.250 align:middle line:84% But it lets you turn these things 00:40:38.250 --> 00:40:40.120 align:middle line:90% into differential equations. 00:40:40.120 --> 00:40:42.960 align:middle line:84% But what you're really doing is just taking the gradient 00:40:42.960 --> 00:40:44.400 align:middle line:90% and setting it equal to 0. 00:40:44.400 --> 00:40:47.910 align:middle line:84% And it just you have to understand functions 00:40:47.910 --> 00:40:49.110 align:middle line:90% are a kind of vector. 00:40:49.110 --> 00:40:51.390 align:middle line:84% I can still talk about linear operators on functions. 00:40:51.390 --> 00:40:53.880 align:middle line:84% I can talk about dot products of functions. 00:40:53.880 --> 00:40:58.350 align:middle line:84% So this linear operator is exactly for the-- 00:40:58.350 --> 00:41:01.560 align:middle line:84% the derivative just is the linearized change 00:41:01.560 --> 00:41:03.430 align:middle line:90% in the output in this integral. 00:41:03.430 --> 00:41:05.130 align:middle line:84% And if I do a little bit of algebra 00:41:05.130 --> 00:41:06.630 align:middle line:84% with the integration by parts, I can 00:41:06.630 --> 00:41:09.630 align:middle line:84% write that as an integral of something times du. 00:41:09.630 --> 00:41:12.760 align:middle line:84% That's a dot product in a generalized sense. 00:41:12.760 --> 00:41:18.270 align:middle line:84% So this thing is a generalized gradient 00:41:18.270 --> 00:41:23.560 align:middle line:84% that you set equal to 0 if you want a minimum or maximum. 00:41:23.560 --> 00:41:26.530 align:middle line:90% Any questions? 00:41:26.530 --> 00:41:30.910 align:middle line:84% And I'll put up some references as well on both 00:41:30.910 --> 00:41:33.310 align:middle line:90% what Frank talked about-- 00:41:33.310 --> 00:41:37.450 align:middle line:84% we have some-- there's some papers, and other reviews, 00:41:37.450 --> 00:41:40.300 align:middle line:84% and things on I-joint methods for ODEs. 00:41:40.300 --> 00:41:43.600 align:middle line:84% And there's lots of papers and books 00:41:43.600 --> 00:41:46.240 align:middle line:90% on Euler-Lagrange equations. 00:41:46.240 --> 00:41:49.090 align:middle line:84% And the traditional topic for deriving 00:41:49.090 --> 00:41:52.750 align:middle line:84% these Euler-Lagrange equations is calculus of variations. 00:41:52.750 --> 00:41:57.160 align:middle line:84% That's what it's called, or variational calculus. 00:41:57.160 --> 00:41:59.163 align:middle line:84% But it really is just taking that-- 00:41:59.163 --> 00:42:01.330 align:middle line:84% now that we've generalized the notion of derivative, 00:42:01.330 --> 00:42:04.040 align:middle line:90% it's the same notion. 00:42:04.040 --> 00:42:05.900 align:middle line:90% It's not nothing really new. 00:42:05.900 --> 00:42:07.720 align:middle line:84% It's just applied to a new vector space. 00:42:07.720 --> 00:42:10.720 align:middle line:84% There's a beautiful, beautiful book by Gelfand and Shilov 00:42:10.720 --> 00:42:13.390 align:middle line:84% that I'll put up a link to on calculus of variations. 00:42:13.390 --> 00:42:14.470 align:middle line:90% I really like. 00:42:14.470 --> 00:42:16.390 align:middle line:84% It's one of those things that starts out-- 00:42:16.390 --> 00:42:19.420 align:middle line:84% Chapter 1 is accessible to a freshman. 00:42:19.420 --> 00:42:22.130 align:middle line:84% And then it gets more and more advanced as it goes along. 00:42:22.130 --> 00:42:26.620 align:middle line:84% But it's a very, very slim book, like 150 pages 00:42:26.620 --> 00:42:28.710 align:middle line:90% or something like that. 00:42:28.710 --> 00:42:32.000 align:middle line:90%