Stokes' Theorem on Manifolds

[Music] you're looking at a picture of the most profound theorem in modern mathematics it's the fundamental theorem of calculus in higher dimensions it's called Stokes's theorem on manifolds and literally all of calculus is a consequence of it here's what it says the total change on the outside equals the sum of the little changes on the inside this is admittedly a little vague so an example might help suppose you have a function f of X and you ask how much does the function increase from A to B Stokes's theorem gives you an answer the total change from A to B is the sum of the little changes between a and B what's a little change in a function the derivative so if you add up all the derivatives of all the points on the inside you get the total change on the outside some of you may recognize this as the fundamental theorem of calculus but it's actually Stokes's theorem in one dimension now let's go to two dimensions what you see right now is something called a vector field I've attached an arrow to every point in two-dimensional space if it helps you can imagine that the arrows describe the flow of water across the page I've placed a blob on top of the vector field suppose I had a little sailboat and a sailed along the boundary counterclockwise I want to know what is the net force that the water applies on in some spots like here the water would flow against me and other spots like here the water would flow with me what is the overall effect Stokes's theorem answers this the overall change on the outside is the sum of the little changes on the inside and in this context change just means spin to see the little spins on the inside you just cut the blob into little pieces each with its own spin but notice something over here one arrow is going up and the other is going down so they cancel each other out same thing here one arrow goes left and the other goes right so they cancel each other out if you do this for all the spins on the inside what are you left with the total spin on the outside some of you may recognize this as greens theorem I don't want to write the whole equation because it's a little complicated here it is if you don't believe me the upshot this is Stokes's theorem in two dimensions the 2d fundamental theorem of calculus so what Stokes's theorem in three dimensions same deal here's a 3d vector field now let's place a blob in this vector field I want to know how much air is flowing into the blob just apply Stokes's theorem in the 1d case we considered the change in the 2d case we consider the spin in the 3-d case we consider the flow so take the region and cut it into pieces each piece has its own flow as before the insides cancel out leaving the outside this result is yet another staple of vector calculus the divergence theorem but actually it's Stokes's theorem in three dimensions now we're ready to discover Stokes's theorem in n dimensions to do that all we have to do is generalize what we saw in lower dimensions to higher dimensions in one dimension we had an interval in two dimensions we had a block in three dimensions we had a region in higher dimension we have a manifold a manifold is a curved surface living in higher dimensional space we can't visualize these things directly so we draw them as if they're 3d next we generalize what we mean by change in one dimension we use to the derivative in two dimensions we used the spin and three dimensions we use to the flow in n dimensions we have the exterior derivative D this is the true derivative it has all these other things as special cases finally we generalize the thing that were integrating first we had a function or a scalar field then we had a 2d vector field and then a 3d vector view now we have an N dimensional tensor field here's the generalized Stokes's theorem the total change on the outside is the sum of the little changes on the inside suppose F is a function and M is a manifold to get the left side we add up F along the boundary of M but that's just the integral of F over the boundary of M the curly D is our symbol for boundary or what about the right side we add up the changes in this that is the exterior derivative of F along M and that's it that's the generalised Stokes's theorem the fundamental theorem of calculus in n dimensions this theorem clears up a huge misconception that people have about calculus that derivatives and integrals are opposites that's wrong people only think that because the fundamental theorem of calculus is written like this in reality we should be writing it like this the integral hasn't disappeared at all the only thing that's changed is the thing that we're integrating over and now we see the truth about calculus derivatives and integrals aren't opposites derivatives and boundaries or opposites now that seems a little silly the derivative is a very algebraic click but the boundary is a very geometric thing but that's what SOSUS theorem says removing the derivative is the same as taking the boundary and removing the boundary is the same as taking the derivative that's why it's called the exterior derivative much like how the boundary is the exterior of a shape and for goodness sakes the symbols even look the same but some of you may not find a satisfying is there an intuitive reason why derivatives and boundaries have anything to do with each other yes the derivative is a local property it zooms in and tells you how a function changes the boundary is a global property it tells you how something changed over all the derivative is a magnifying glass the boundary is a big picture of you but they are both the same thing Martin Fisher the famous American physicists once said knowledge is a process of piling up facts wisdom lies in their simplification this is the moral of Stokes's theorem it brings all of calculus into one unified framework Albert Einstein said it like this genius is making complex ideas simple you

Channel: Aleph 0

Views: 174,543

Keywords: calculus, math, dimension, topology, manifold, stokes, theorem, curl, fundamental theorem of calculus, divergence, differential geometry, generalized stokes theorem

Id: 1lGM5DEdMaw

Length: 6min 19sec (379 seconds)

Published: Sun May 03 2020