The Core Idea Connecting the Biggest Theorems in Calculus

The biggest theorems in calculus are connected by one fundamental idea. I'm going to tell this story over two videos. And in this first one, I'm going to call it advanced mountain climbing using calculus. But more precisely, I will illustrate my favorite perspective for understanding something called the fundamental theorem of line integrals. Let's start by climbing a two-dimensional mountain to build our intuition. Then we'll upgrade. This mountain is described by a function y= to f ofx. And suppose my goal is to understand the net change in height as I climb the mountain starting at A and finishing at B, which is just the difference in the heights of the end points f of B minus F of A. Climbing a mountain takes a bunch of small steps. And in calculus, we often convert a big problem into increasingly large numbers of smaller problems that are hopefully easier to solve. So, if I approximate my curve with a lot of small steps, notice how adding up all the heights of the little steps is the same thing as the big net change in height. Let's zoom in on one of those little steps. I'll say it has a width delta x and a height delta y, where the symbol delta helps to remind us that we're imagining a small change in x or y. How can we estimate that little change y in the height of our step? The big idea of differentiation is that we can compute the slope of a tangent line at a point. And this allows us to estimate the value of delta y as approximately f prime of x * delta x. I like to think of derivatives as a scaling factor, if you will, between the input x and the output y. So that's one step. And if we add up all the steps, I use the symbol sigma here to denote adding them all up. What we get is the idea that the sum of all of the little changes in heights for all of the steps would add up to be the big net change in height. And while this is an approximation that breaks up our interval into a finite number of little steps, the magic of the fundamental theorem of calculus is to break up the path into infinitely many infantesimally small steps. Our notation then changes. Here we use an integral sign which I want you to think of as just meaning I'm adding a bunch of things up. And then what we're adding is all of those heights for all of those little steps where we're now using dx not delta x to emphasize notationally that these are infantesimally small. So ultimately we have sketched the intuition behind the fundamental theorem of calculus number two which has this interpretation that the sum of all the little changes add up to the net change. Now calculus students everywhere use this theorem all the time often focusing more on its computational side. If you wanted to integrate a function like 2x from 1 to 3 for instance you find a function like x^2 whose derivative is 2x you plug in the three. one into that x^2, you subtract, and you evaluate the integral. Hopefully, I've added some geometric flavor to that fundamental theorem. All right, so that's 2D

mountains. Now, let's upgrade to three-dimensional mountains, or in other words, functions that we can visualize as the height being a function of x and y. If we have some path that climbs that mountain, we might again be interested in computing the net change from my initial height to my final height. I'll once again break up my path as a little staircase. And the big net change can once again be thought of as the sum of all of the little changes in height. But what are those little changes? Before we approximated them using the single variable derivative. So what is the analog of that in the higher dimensional setting? This takes a bit of work or perhaps I should say a bit of play. Imagine I'm standing at a point and I look in a specific direction. I can imagine walking straight in that direction and this gives me a curve along the mountain. As I change my direction, the curve changes. So how steep my path is at any given point can be represented once again with tangent lines. And the slope of that tangent line tells me how steep my path is in a specific direction. The fancy name for the slope of these tangent lines when looking in a specific direction is called a directional derivative. I. e. the derivative in a particular direction. But how can we compute it? In our mountain climbing context, the direction we are traveling is given by a

specific path that we're on. And the direction that we're pointing changes as we go along that path. And I can draw the path in the two-dimensional domain. And our notation will be to call it the path r of t. So this is a parameterized path from some time interval to locations r of t on the xy plane. For example, the path given by cosine of t sin of t for t values between 0 and pi gives this semic-ircular path. If I want to know what direction I'm heading at any point in my curve, I can compute r prime of t. that is the derivative of the path. It's easy to compute like in our cosine t sin of t example. r prime of t is the derivative of each component. So minus s of t and cosine of t. So that's one relevant vector to our story. The direction that came from the tangent vector r prime of t to our path.

But to address our original question about directional derivatives, that is the steepness in any specific direction, I also need some information about the height function for my mountain. Let's introduce the height information in the two nice directions first. When I'm pointing either parallel to the x-axis or parallel to the y-axis, that is the nice directions in terms of the cartisian xycoordinate system that we are working with. If I look parallel to the x-axis, this means that y isn't changing. And so the directional derivative in this direction is just called the partial derivative with respect to x. We can compute this because if your function is something like x^2 y cubed because y is constant, the derivative with respect to x is just the 2x and then multiplied by y cubed. That part of it doesn't change. it's held constant. Similarly, if you look parallel to the y ais, you get the partial derivative with respect to y. And this tells you the slope in the y direction. And it's computed similarly, holding x constant. And so its partial derivative with respect to y is x^2 * 3 y^2. You can put these two partial derivatives together, the partial of f with respect to x first and then y second into something called the gradient vector at any point. And I'll denote this with this upside down triangle called NAB. The gradient vectors have a really lovely geometric meaning. Here I've drawn on my mountain a bunch of level curves which are curves of constant height. That is if you walk along a level curve, you don't increase your height on the mountain. Now I'll plot the gradient vectors at a whole bunch of points in the xy plane. I notice first that they all seem to be pointing up the mountain, i. e. the direction that you should walk if you wanted to climb straight up the mountain as fast as possible. And the vectors are longer the steeper we are on the mountain. If I rotate my perspective to be looking straight down, you can see more clearly that the gradient vectors are orthogonal to those level curves. Walking along a level curve doesn't change your height, but walking orthogonally to level curves increases your height as fast as possible. So the gradient vectors collect that information about how the height function is changing that we are interested in. Now we've understood the two key vectors for our big result. The gradient vector is going to tell us the steepest possible direction and the vector r prime of t tells us for our specific path which direction are we actually going. So what I can do is take the dotproduct of the gradient vector and the rp prime of t and dotproduct is this operation that measures the alignment between two vectors. So for instance if we were walking along a level curve where our height wasn't changing then the r prime of t would be tangent to that level curve and it would be orthogonal to the gradient vector. So our directional derivative would be zero no change in height. But the more our path aligns with the gradient vector, the closer to that maximum steepness up the mountain we're going to be traveling. Let's return to our big goal.

We were walking up the mountain along a path and we had a lot of little steps along the way. The way we're going to now approximate all those little changes in height is via that gradient of f dotted with rp prime of t. That will be my stretching factor of a little change in time delta t. This is the analog of frime of x delta x that we saw in the lower dimensional case. We've just replaced that stretching factor of frime of x with the more higher dimensional analog of the gradient of f dotted with the rp prime of t to describe how much height is increasing for each step as we change an amount delta t along the path. So adding up all the little changes in height once again represents the net change in height. And so our calculation is effectively looking at the gradient vector and the tangent vector at each time step taking their dotproducts and then adding them all up. And when we do this breaking it into infinitely many infantesimally small pieces, we rephrase it using the integral sign to replace sums and the dt to replace the deltat t. This equation gives us the fundamental theorem of line integrals and its colloquial description is the same as before. If you are to add up all of the little changes in height, you get the net change in height over your entire path. And really, this is a generalization of the fundamental theorem of calculus. If your path is just going along the interval from a to b in the x axis, then our gradient of f dotted rp prime of t is just the same thing as frime of tdt. So the same formula as before with perhaps some reabeling of x to be t. One really cool example is that the net change in height is actually independent of the path that you take. If you and I take different paths around the mountain, but as long as we start at the same place and end up at the same place, our net change in height should be identical. So this path independence property is very interesting and it begs the question of what other types of integrals share this kind of path independence. In fact, normally this fundamental theorem of line integrals is presented in the context of vector calculus where one first starts with an arbitrary vector field and asks about path independence when integrating along a path the tendency for the path to be aligned to the vector field. I've previously done videos on this standard approach. I'll link to it below and I'll also link to a video showing an example computing the quantities from this video. In the next video, our regions are going to be more interesting than just paths. We'll consider surfaces and threedimensional volumes and our notion of change will be more interesting as well and will lead us to some of the biggest theorems of vector calculus. Green's theorem, Stokes's theorem, divergence theorem. And all of those are unified by the idea from this video that when you accumulate small changes over a region, you end up with the net change on the boundary of that region. Now, I wanted to share with you one of my favorite ways to learn math and science, which is brilliant. orgs. or all the animations from this video were recorded with Python. But a year ago, I didn't know any Python and so I went to Brilliant introduction to Python and I actually worked my entire way through their sequence and I found their courses really effective. Everything was really interactive like I got to test my own understanding, manipulate the code, see what the output was. And I found that the process of building up the skill hierarchy layer by layer to be really effective at having both long-term retention and procedural fluency for me as a learner. If you enjoyed today's video, you might be particularly interested in their calculus course. And I really love that their calculus course was highly visual to the point that you actually get to manipulate those visualizations and that helped to build up a lot of the underlying understanding and sort of conceptual reasoning behind what's going on in calculus. Whether you're 10 or 110, as long as you're interested in learning, Brilliant does a great job of supporting that learning. To learn for free on Brilliant for a full 30 days, go to brilliant. org/drebbazet. Scan the QR code on screen or click the link down in the description. Brilliant. Also give viewers of this channel 20% off an annual premium subscription. With that said and done, I hope you enjoyed this video. If you have any questions, leave them down in the comments below, and we'll do some more math in the next