Why you don't understand GREEN'S THEOREM -- Geometric Algebra, Calculus 3, Vector Calculus

[Music] so in today's video what we want to look at is in my opinion the right way to view green's theorem so green's theorem is this relation on line integrals that we learn about in any kindergarten vector calculus course it's very confusing for students it's not clear at all why there's these minus signs why you can express a line integral over a curve in terms of a double integral over the region it bounds all of these theorems like stoke's theorem divergence theorem all this stuff it's just a whole bunch of ad hoc nonsense that you just walk away at the end without having any idea what's going on to anyone who has any knowledge of what's referred to as durham core homology vector calculus course is just one one or two facts repeated over and over again the purpose of this video is to illustrate or at least give you some idea of this higher perspective so what we're going to do is we're going to develop the right language to talk about these theorems in vector calculus namely we're going to talk about the wedge product and exterior derivative and things like that you won't have to worry about the jargon necessarily but you'll at least see why things are occurring and give a and we'll give a clean presentation of this material okay that's enough chit chat let's get into it okay so let's recall the statement of green's theorem recall that green's theorem says that the line integral over the closed loop c of p d x plus q d y where p and q are functions of two variables is given by the double integral over the region at bounds which we've denoted by d of partial q partial x minus partial p partial y of dx d y now to have some intuition and actually understand what green's theorem is saying let's recall the fundamental theorem of calculus that we learned in kindergarten so the formulation i'm going to take here is that the integral of a to b of f prime of x dx is equal to f of b minus f of a so if we look at the graph of f prime of x the fundamental theorem of calculus tells us that if we take two points a and b then the area underneath this graph is determined by the value of another function namely f at these boundary points alone now a priori we need to know every point between a and b to determine the area but the fundamental theorem of calculus tells us that it's actually only determined by the values of a separate function namely its antiderivative at the endpoints now to get further insight on the fundamental theorem of calculus what we'll do is we'll rewrite the integral from a to b of f prime of x dx as the integral over the closed interval a b f prime of x dx for us this is going to be the same and what we can do is we can also write f prime of x as df dx and what we can also do is write f prime of x as df and we can also write a b as the boundary of the closed interval a b so to denote the boundary of a set we use this partial symbol so the set containing only the elements a b is denoted by partial of the closed interval a b so now what the fundamental theorem of calculus says is that the integral over the closed interval a b of d f dx is equal to the integral over the boundary of the closed interval of a b of f dx now just to make this subsequent remark or transparent what we can do is it can actually replace df with partial f since we're only dealing in the one variable situation this is all equivalent and so we've written the fundamental theorem of calculus as the integral over the closed interval a b of partial f dx is equal to partial of the closed set a b in other words the boundary of the closed interval a b of f dx from this way that we've written the fundamental theorem of calculus it's clear that integrals are not necessarily the opposite of derivatives what's happening is that the opposite of the derivative is actually the boundary of the region of integration so the way in which this discussion that we've had so far is formalized is through what's referred to as durable corphomology this is a subject within differential geometry that is covered within any standard text on romanian geometry or differential geometry or being it's quite an advanced concept subject to typically a graduate course the this can be found for example in jost's romanian geometry and geometric analysis or my favorite book on the subject which is peter peterson's romanian geometry the second edition these are excellent books which i would highly recommend i initially learned from money in geometry from jost's book under ben andrews and then as a graduate student i worked through and continued to work through peter peterson's reminding geometry book to continue this extension of the fundamental theory of calculus you know hope to get some insight on green's theorem what we'll now do is ask the question of how can we formulate area the notion of area this abstract idea into an operation that we can perform on vectors or something or some type of object we're familiar with in order to get a language that's actually fruitful so we want to develop a language on vectors or on some type of objects that furnish an understanding of what area is so let's restrict our consideration to vectors so let's consider the vector v which is the vector 1 2 and the vector u which is given by 2y what we can of course do is translate v to the end point of u and u to the endpoint of v and form this parallelogram now of course from kindergarten linear algebra we know that the area of this parallelogram is just going to be given by the determinant of the matrix whose columns are given by u and v but we'd like to express that in an operation but we'd like to know whether we can actually construct an operation on vectors such that if we take a vector u take a vector v apply this operation it gives us the area of the parallelogram this will be an important key step in the development of this formalism so let's assume we have such an operation we'll denote it by this wedge symbol here so by u wedge v we're going to mean the area of the parallelogram formed by u and v to get an understanding of exactly what properties this operation would have what we'll do is we'll write u as a i plus b j and v as c i plus d j now the wedge of u and v is then just a i b j wedge c i plus dj we'll assume that it's distributive so we can foil it out in the sense that we'll now have a i wedge ci plus dj plus bj wedge ci plus dj and then this would further expand to a c i wedge i plus a d i wedge j plus b c j wedge i plus b d j wedge j now of course i wedge i is the area of the parallelogram formed with i in itself and j where to j is the area of the parallelogram formed with j in itself that's just the area of a line which of course is zero so now what we can do from what we've just calculated is that in this u wedge v calculation the first term and the last term which are the i wedge i and j wedge j terms are both zero what we're now left with therefore is that u wedge v is a d i wedge j plus b c j wedge i and now we want to understand how does this wedge operation behave if we switch the order so what is the difference between i wedge j and j wedge i so we'll recall that i wedge i and j wedge j well they're areas of lines so they're going to be zero and in particular i can consider the parallelogram formed with i plus j and itself again that's just going to be the area of a line so it will also be zero so we have that zero is equal to i plus j which i plus j expanding this out we get i wedge i plus j plus j wedge i plus j which we further expand to i wedge i plus i wedge j plus j wedge i plus j wedge j is exactly what we did before again the first term and the last term will cancel and we see that 0 is equal to i wedge j plus j wedge i in other words we get the following anti-symmetric property which is that i wedge j is equal to minus j wedge i so when we switch the order with respect to this wedge operation we pick up a minus sign so now if we return to our calculation here namely the u wedge v is a d i wedge j plus b c j wedge i then switching the order expressing everything in terms of i wedge j we get a d i which j minus b c i wedge j which of course we can just write as a d minus b c i wedge j which may surprise you or may not but that's exactly what the determinant of a two by two matrix is if the matrix is a b c d so this wedge operation is exactly what we wanted it gives us the area of the parallelogram of the vectors where the vectors are columns of this matrix so this wedge operation gives us exactly what we wanted we want the area of the parallelogram formed by the vectors u and v now this operation on vectors is well known it's it's known as the wedge product so the defining properties of the wedge product are that if we wedge something with itself we get zero again that's because it's the area of a line a parallelogram formed by a vector in itself is going to be a line with area zero and it has the following anti-symmetry property which is that u wedge v is minus v wedge u okay so let's see how this new language tells us anything new or gives us any insight into green's theorem so recall that green's theorem again is the line integral over a closed loop c of p d x plus q d y that's the double integral over the region it bounds of partial q partial x minus partial p partial y d x d y what we're going to do is we're going to write omega as p d x plus q d y and we want to compute the derivative of omega which we denote by d omega now that's going to be the derivative of the first function dx plus the derivative of the second function dq dy now we're dealing with a function of two variables and we're going to use the following definition of the derivative which is that if we have some function f then d f is partial f partial x dx plus partial f partial y d y this is well known to be what's referred to as the exterior derivative but the name is not important here so if we use this definition of the derivative what we see is that dp is going to be partial p partial x dx plus partial p partial y d y and then we have the dx plus now dq is partial q partial x dx plus partial q partial y d y and then we'll have d y at the end now what's implicit here is that they're actually going to be wedges here so whenever we write dx dy or dx dx implicit in that is that there is a wedge these are referred to as co vectors which can just be thought of as vectors for this discussion and we're pairing them in such a way that we get area that's what integrals yield so what we have is partial p partial x dx wedge dx plus partial p partial y d y wedge dx then similarly partial q partial x dx wedge d y plus partial q partial y d y wedge d y this is just distributing the previous line now recall that the wedge product has the property that if we wedge something with itself that will give us the area of a line and that will be zero so immediately we see that the first term and the last term here partial p partial x and partial q partial y terms both vanish and we're left with partial p partial y d y wedge d x plus partial q partial x d x wedge d y now the other property of the wedge product was the anti-symmetry so if we switched the order of what we were wedging we would pick up a minus sign in particular what we see is that what we have here is minus partial p partial y dx wedge d y plus partial q partial x dx wedge d y and then of course we can combine this and just write this as partial q partial x minus partial p partial y of dx wedge d y and if we compare this with green's theorem what we see is that the left hand side is the integral over the boundary of d so we can write c as partial d of omega and we can write the right hand side namely this double integral as the integral over the entire region you can keep the double integral notation if you'd prefer of d omega and what this right hand side of green's theorem is saying is that we have the integral over the entire region of the derivative and so in particular green's theorem is saying that the integral over the boundary of omega is the integral over the entire region of d omega when we write it like this symbolically we're only referring to regions in the plane but it holds in higher dimensions this is in its full generality it's referred to as stokes theorem what i hope has been made clear at this point now is that green's theorem when expressed in this language of wedge products and in my opinion expressed in the right language is nothing more than the fundamental theorem of calculus now the fundamental theorem of calculus which in its great generality is stoke's theorem is a deep theorem it's highly non-trivial and in fact when i was an undergraduate learning this stuff i remember that one of the lecturers said that you know you're allowed to have your favorite theorem you can your every every mathematician gets the right to choose their favorite theorem whatever theorem is most significant to them but no mathematician gets to choose their second favorite theorem because that's stoke's theorem stokes theorem has so many applications it's it's so important in in a field of mathematics and it's called homology and chromology is really one of the one of the big machines in mathematics today if you'd like to see how actual chromology can be used in very concrete ways without getting bogged down in all this jargon i'd suggest looking at the series of videos i made on wheeler's number where the euler's number emits a geometric representation in the sense of can can it be formalized in a geometric way can it be given a geometric interpretation i'll have the link to those videos in the description down below you

Channel: Kyle Broder

Views: 55,354

Keywords: University Mathematics, geometric algebra, mathematics degree, Geometric Algebra, Derivative in Geometric Algebra, Derivative in Exterior Algebra, derivative of vector field, Vector Calculus Derivatives, mathematics, Fundamental Theorem of Calculus, Stokes theorem, Green's theorem, Mathematics degree, Mathematics major, mathematics for machine learning, mathematics career, mathematics definition, mathematics degree jobs, mathematics with computer science, mathematics formula

Id: XEDtm09ol5o

Length: 16min 17sec (977 seconds)

Published: Mon Jul 05 2021