The Essence of Multivariable Calculus | #SoME3

this is me in my first year of college I took multi-variable calculus of course that many of you may be familiar with when I was taking the class my roommate let's call him Ari was a physics and CS double major he had taken the course before so he offered me these words of wisdom the way that multivariable calculus all comes together at the end is just amazing now let me tell you even after taking the class I had absolutely no clue what Ari was talking about despite all of the long nights of double integration and essay writing true story we wrote essays for calculus I still didn't get it the problem was I understood all of the theorems and Concepts individually but I couldn't see how they were all related and I think that's a boat that a lot of students are in seeing connections between things is all about perspective I can show you thousands of pictures of the Earth but you'll never gain the perspective that astronauts have when they look down on our planet from thousands of miles in space what Ari was describing this idea that multivariable calculus all comes together somehow was a perspective and in this video I'm going to present that perspective to you by building multi-variable calculus from the ground up so let's get into it I'm going to talk about five different theorems here the fundamental theorem of single variable calculus the fundamental theorem of line integrals Green's theorem stokes's theorem and the Divergence Theorem theorems 1 and 2 are very similar three is actually a special case of four and by the end of the video you'll see how all five of them are really the same thing as we build up from one theorem to the next we'll work with higher dimensional objects and spaces as a disclaimer I'm not going to spend much time rigorously proving these theorems it's not important for this video since we just want to see how they're all related let's start with the fundamental theorem of single variable calculus the formula for this theorem can be written like so what this is saying is that the difference between the value of a function f at x equals B and its value at x equals a is equal to the integral of its derivative from the balance x equals a to x equals B to build up calculus from the ground up I'm going to break this formula down step by step we can write the derivative of f in prime notation as shown or in leibness notation like this leibniz notation essentially writes the derivative as a quotient DF divided by DX where DF is a very very small change in F and DX is a very very small change in x mathematically derivatives are not actually quotients but writing it this way can help us understand the geometric intuition behind this formula let's consider the graph of f versus X here I just chose F to be a random curve that is both continuous and differentiable along the interval a comma B what that means is relevant for proving this theorem but not too important for our intuitive explanation if we look back at our theorem in liveness notation we can sort of pseudo cancel the DX's to get just the integral of DF from A to B mathematically you can't actually do this DF DX is not a quotient but an operator DDX acting on F but we'll write our formula like this just to help develop intuition remember that DF is a very very small change in the value of the function f antesimally small in fact graphically this would correspond to a vertical movement from one value of f to another so we understand what DF is now let's check out the rest of this truncated formula integrals can also be thought of as infinite sums in our case this integral means we're adding up an infinite number of infinitesimally small differential components DF so this integral is really telling us to add up all of the infinitesimally small changes in F from x equals a to x equals B the sum of these tiny changes in F from A to B is just the total change of f from A to B and so we can see that the integral of DF from A to B is the same as the difference between the value of the function at B and its value at a now let's rewrite our theorem the way it was originally written with 5 minutes notation just as DF was an infinitely Small Change in F the x is an infinitely Small Change in X which would be a very small step in the horizontal Direction on our graph you might have heard that DF DX is the slope of a tangent line to F at a given X this makes sense DF is our vertical rise and DX is our horizontal run and if we divide rise over run we get the slope to get DF we just have to multiply that slope by the corresponding DX at a given point so really when we're taking an integral we're summing all of the products of the slopes times their infinitesimal runs from A to B which is the same as summing all of the infinitesimal rises from A to B the reason we have to consider each infinitesimal DX is that the slope dfdx changes slightly at each value of x when we describe the fundamental theorem in this way we've essentially broken down the ideas of integration and derivation completely integration is the infinite summation of infinitely small components while derivatives are simply slopes multiplying a slope by an infinitely small run gives us an infinitely small rise and taking the infinite sum of all the Rises gives us the total rise and that total rise is the net change of our function I'll say it one more time just for emphasis multiplying a slope by an infinitely small run gives us an infinitely small rise and taking the infinite sum of all those Rises gives us the total rise and that total rise is the net change of our function so that was a lengthy intuitive description of the fundamental theorem of single variable calculus we can of course make this description even more concise and that will be of great use to us in understanding how multivariable calculus all comes together but it also means we have to make a sacrifice unfortunately in higher Dimensions this simplified description of integrals and derivatives doesn't quite hold up our derivatives aren't always going to be slopes and our integrals aren't always going to be infinite sums along axes but instead infinite sums over new weirder shapes we'll get into what that looks like but for now this just means that it's better for us to describe the fundamental theorem of single variable calculus not with the dimensionally restricted concepts of slopes and sums but with the more abstract ideas of derivatives and integrals so with that let's put this formula into words the integral of the derivative across the bounds is equal to the original function at the bounds keep the sentence in the back of your mind we'll put it aside just for a moment in our formula dictionary a set of conversions of calculus formulas to plain English we'll continue to add to this dictionary over the course of the video and get back to it near the end to answer our overarching question now let's discuss line integrals from the name they suggest we are taking the integral over a line but what exactly does that look like well for our purposes we're going to be looking at line integrals of vector fields you can think of a vector as an arrow with a magnitude or size and a Direction Vector fields are kind of like functions that output a vector to each point in Space the line integral of a vector field over a curve C is calculated in a similar way to the integral of a regular function for the 1D case we multiply something in the fundamental theorem that something was a slope of the tangent line by tiny changes in X to get a tiny change in y and add up all those tiny changes in y to get the overall change in y for the case of our Vector field our tiny changes in X are now the vector derivative of our curve c c is a vector-valued function written as R of T which is similar to a vector field except now each point on the curve C is defined by a vector that points in the origin to a point on C because C is Vector valued it defines the X and Y coordinates of its points in terms of a new variable T so we have derivatives in both the X and Y directions which together sort of make a derivative Vector R Prime of T we take this derivative vector and multiply it by the vector in our Vector field at a given point except we can't multiply two vectors so instead we take their dot product adding up all of these dot products at different points along a curve C gives us the final value of our line integral mathematically this idea can be written like so our something is now a vector field our tiny changes in X are now derivative vectors of C our tiny changes in y are now dot products and instead of adding all the tiny changes in y we add up all the dot products to get our overall line integral the bounds A and B are now the endpoints of our curve C the X and Y points inputted into our Vector field are often just written as R of T because those are the points on our curve C oftentimes we can also truncate this derivative Vector to just Dr to better represent the idea that this is a derivative vector and finally we usually don't write the bounds over integral As A and B but instead just write C at the bottom indicating to take the integral over the whole curve C if this description was confusing that's because I glossed over a few important details but don't worry about that for now I'll make a video later explaining line integrals in much more depth for now let's take this idea of taking the integral over an arbitrary curve for granted and examine the bigger picture the reason we care about line integrals of vector Fields is that the gradient of a regular multivariable function is a vector field well then what the heck is the gradient when we have a function with multiple inputs a multivariable function we don't just have one derivative we have many partial derivatives derivatives of the function with respect to each input variable assuming all other inputs are constant and when we compile all of these partial derivatives as a vector to make a vector field we get the gradient the gradient points in the direction of greatest change of the value of our function you can think of each of these partial derivatives as moving along the axis of that respective input variable this would be moving along the x-axis this along the y-axis this along the z-axis and this along the W axis if we have too many input variables it becomes impossible to visualize so for demonstration purposes let's stick with just two input variables and one output this way we can graph our function in 3D space and we can graph its gradient in 2D space let's use the function f of x y is equal to x squared plus y squared to demonstrate gradients if we take the gradient of this function we get the following Vector field where each Vector in space is defined by twice its horizontal x coordinate and twice its vertical y coordinate now let's draw some random curve C in our gradient field and try to find its line integral let's make this curve the line segment of y equals x squared going from one comma 1 to 2 comma four once we take the line integral of our gradient field over this curve C and the details of this are not particularly important we find that our line integral is 18. interestingly if we plug in the endpoints of our curve c 2 comma four and one comma 1 into our original multivariable function f we get 20 and 2. and their difference is the same as our line integral this ladies and gentlemen is a fundamental theorem of line integrals where it's informally the theorem looks like this let's put this equation into words just as we did with the fundamental theorem of single variable calculus the integral of a gradient over the integrating curve is equal to the original function at the curves endpoints let's add this to our formulas dictionary for the last three theorems I'm not going to go into as much detail on the specifics as I did for the previous two there are plenty of incredible videos out there visualizing these theorems in different ways instead I'm going to focus on the bigger picture so we can get an answer to our initial question how does multivariable calculus all come together now let's talk about Green's theorem the formula for this theorem looks like this looking at this for the first time it can be difficult to understand exactly what's going on here especially if you're not familiar with double integrals let's break it down for this theorem we're still just working with the 2D X Y plane on the left side we have a line integral over a curve C just like before except now instead of finding a DOT product of a vector field and a derivative Vector we multiply each component of the vector field by the differential component in its respective Direction mathematically it's the same operation just written differently we multiply the horizontal part of our Vector field by a differential movement in the horizontal Direction along c and the vertical part of our Vector field y differential movement in the vertical Direction along c then we add up all those products over the entire curve C this circle here means that our curve is closed and that we want to take the line integral over the whole closed curve C on the right side we have a double integral over an area D this area is enclosed by our curve C and we're taking the integral of this weird combination of partial derivatives it turns out that this combination of partials can be interpreted as a measure of how much a vector field curves or curls you can think of this as a tiny Circle Vector existing on a differential area component like this all the double integral says is that we need all of these tiny curls times the tiny area component they are on in both the X and Y directions separately which is what it means to integrate over an area it's basically just two consecutive integrals first take the integral with respect to X get some result and then take the integral of the result with respect to y the conceptual idea behind Green's theorem is that adding up all of these little Circles of a vector field inside an area is the same as taking the total curvature of the vector field along the shape's boundary going back to the raw equation we can put it into words just as we did with the other two theorems the integral of the integral of the derivative over the integrating area is equal to the integral of the original function over the boundary curve let's store this statement away in our formulas dictionary as well moving on to stokes's theorem the equation is written like this notice that the line integral on the left side of the equation is exactly the same as a line integral in Green's theorem just written differently the right side is still a double integral but now we're taking the double integral of something else graphically we're still dealing with a 2d shape enclosed by a 1D line except now our 2D shape can live in 3D space if we so choose making it a surface rather than an area this weird value inside the double integral is actually called the curl the curl is essentially another form of a derivative like the gradient it involves partial derivatives except this time it's of a vector field and we're multiplying those partials in a weird way again the details of how we calculate curl are not important what is important is understanding what curl is intuitively curl is actually a vector field if we have a portion of our initial Vector field in 3D space that curls around like this the curl Vector points perpendicular to our Vector field and its magnitude measures how much our Vector field will curls or Curves around in circles this concept is exactly the same as the weird derivative expression in Green's theorem in fact that expression is equal to the curl of a two-dimensional Vector field so really Green's theorem is a special case of stokes's theorem where our surface only exists in 2D space the other important part of the double integral is this DS component it tells us that we're taking what's called a surface integral an integral over surface the differential components of our surface area are small surface patches and we describe these with vectors of infinitesible magnitude pointing perpendicular to our surface this is DS we take the dot product of these surface vectors and the curl which points perpendicular to our Vector field and sum them all up over the whole surface so altogether stokes's theorem tells us that if we have a 2d surface existing in a 3D Vector field adding up all of the tiny curls of the vector field over that surface is equal to calculating the curve of the vector field over the boundary of that surface let's put this formula into plain English as we did before the integral of the integral of the curl over the integrating surface is equal to the integral of the original function over the boundary curve let's store this away in our formulas dictionary finally we have the Divergence Theorem this theorem is another higher dimensional extension of the theorem of line integrals and Stokes theorem first we dealt with integrals over lines then integrals over surfaces now we're going to look at integrals over volumes the equation for the Divergence Theorem can be written like so on the left side we have a double integral over a surface s on the right side we have a triple integral over a volume e a triple integral is an integral in three dimensions similar to how a double integral was taken in two dimensions visually the volume we are integrating over on the right side is enclosed by the surface written on the left side unlike stokes's theorem instead of taking the double integral of the derivative form we're now taking the double integral of the vector field itself our derivative form is inside the triple integral this derivative form is called the Divergence the Divergence of a vector field is similar to gradient and curl in that it's another way of taking the derivative except it doesn't return a vector field but instead a scalar function it's also much easier to compute the Divergence is just the sum of all the partial derivatives of a vector field conceptually the Divergence measures how much a vector field points towards or away from itself visually this means that if we take an infinitely small cubic volume DV a vector field with a large positive Divergence around that DV would have a lot of vectors pointing out of the volume in all directions so we would call it a vector source a vector field with a large negative Divergence around that DV would have a lot of vectors pointing into the volume in all directions making it a vector sink the right side of this equation is telling us to multiply the Divergence of our Vector field by the infinitely small volume components and then add them all up so really it's giving us the sum of how much the vector field points outward in each tiny volume inside our 3D shape the other side of this equation is a closed surface integral like the regular surface integral we computed in stosis theorem we're taking differential area components on our surface represented by a vector perpendicular to the surface and finding their dot products with the vector field at a given point then adding them all up except this time our 2D surface is closed because it surrounds a fixed volume the 2D surface and Stokes theorem did not enclose any volume in a way the surface integral of a closed surface is a measure of the vectors pointing out of the surface this is also known as the flux so altogether the Divergence Theorem is telling us the sum of all the little pointed outwardnesses of a vector field inside a 3D shape is equal to the total pointed outwardness of the vector field over the boundary of that shape let's put this equation into plain English as well the integral of the integral of the integral of the Divergence over the integrating volume is equal to the integral of the integral of the original function over the boundary surface this statement can also be added to our formulas dictionary in fact let's take out all the statements in our formulas dictionary and analyze them now well this just looks like a lot of words we can use what I like to call the mathematical properties of the English language to simplify these sentences if we see two of the same word or phrase on either side of an is equal to phrase we can cancel them just like we would with a mathematical equation if we look at these last three statements we can cancel one or two integrals on either side of the is equal to so we're left with this now let's generalize some of these nuances here over the integrating curve the integrating area the integrating surface the integrating volume curves areas surfaces and volumes are all things we integrate over and we can generalize them as shapes in the same way the endpoints boundary curve and Boundary surface are all just boundaries so we can simplify them like so and finally we know that the gradient the curl and the Divergence are all different forms of the derivative so we can generalize them to just the derivative and now we see a pattern emerge the integral of the derivative over a given shape is the original function at along or over the shapes boundary this my friends is what connects all of multivariable calculus how it all comes together take a moment to let this sink in and maybe go back to other parts of the video to brush up on the theorems that led us to this idea luckily for us some genius mathematicians have generalized this into a nice and neat formula called the generalized doses theorem written like this the generalized doses theorem states that the integral of a differential form over a boundary the Omega of some orientable manifold Omega is equal to the integral of its exterior derivative over the whole of Omega now I know that was a mouthful but this this is it this one equation lets you generalize all of calculus in any number of Dimensions that you want let's analyze it a bit more thoroughly as a disclaimer I have not taken differential geometry so I'm not really in a place to rigorously Define an orientable manifold or an exterior derivative for you but the core idea in this equation is exactly what I've discussed throughout this whole video you can think of this differential form as just the function or vector field that we are integrating over what's special about this differential form is that well it's differentiable or in other words has an exterior derivative for the fundamental theorem of single variable calculus the exterior derivative is just the normal one-dimensional derivative we think of as a slope for the fundamental theorem of line integrals the exterior derivative is the gradient for greens and stokes's theorems the exterior derivative is the curl and for the Divergence Theorem the exterior derivative is the Divergence Omega here is the shape we integrate over and D Omega is the boundary of that shape for line integrals Omega is the integrating curve and D Omega is its endpoints for greens and Sox's theorems Omega is a Surface we are integrating over and the Omega is the closed boundary curve of that surface and finally for the Divergence Theorem Omega was our integrating volume and D Omega was the surface in closing that volume the single integrals on either side of this formula are really just truncated representations of multiple integrals with the boundary side of the equation always being one integral less than the shape side putting all that together this theorem is the ultimate culmination of the pattern we observed among all the theorems of multivariable calculus the integral of the derivative is equal to the original function along the boundary I'll say that again just for emphasis the integral of the derivative is equal to the original function along the boundary and that's it if you've made it this far congratulations you just have everything you need to know to have the theorems of vector calculus ingrained into your mind and if you still don't get it that's okay too this perspective on multivariable calculus only came to me about six months after the end of my course it happened right outside my college dorm in a car at 1am with three of my friends one of them was stressing about his upcoming multivariable calculus final and so my other friend a senior and another physics and CS major was trying to console in doing so he said something along these lines in higher math all of those theorems the theorem of line integrals stokes's theorem Divergence Theorem they're all the same thing he then proceeded to explain everything I described in this video with a lot of finger motions I mean physicists are quite good with their fingers not that I would know I'm just a foolish chemist the reason that my second physics and CS friend was able to enlight me that night was partly because he had taken higher level math so he had the perspective on calc 3 that stems from the generalizedosis theorem it was this perspective that he shared with me that night that stuck with me forever I mean it's what took me from here to here goes to show that sometimes all you need to solve a problem is to look at it from a different angle to me it's crazy to think about how all five of these complicated theorems can be generalized so simply take it from a guy who doesn't even study math I'm a chemist and I still spent all of this time making a video about math because it was a topic a perspective that I found earth-shatteringly awesome truth be told the generalized stokesis theorem really just reflects one of the greater themes of math the more math you learn the more you realize everything is the same in a way you could say that all of math converges to that one astronaut meme it's all stokes's theorem always has me and so to be honest if you're gonna take away only one thing from this video it shouldn't be that the integral of the derivative is the original function at the boundary it's that you should find yourself a physics and CS friend in college [Music]

Channel: Foolish Chemist

Views: 122,490

Keywords: calculus, math, learn, multivariable calculus, multivariable, what is calculus, introduction to calculus, 3blue1brown, some3, summer of math exposition, green's theorem, stokes' theorem, stokes, stokes theorem, divergence, curl, divergence theorem, fundamental theorem, fundamental theorem of calculus part 2, fundamental theorem of calculus, education, educational videos

Id: Jk_k3q9RoMU

Length: 29min 5sec (1745 seconds)

Published: Thu Aug 17 2023