Green's Theorem, explained visually

In this video we're going to be building up a relation between a double integral and the line integral if You don't remember what a line integral is. I did a video on this here The theorem behind this is called greens theorem Let's say that, we have a 2-dimensional vector field, which I'll call F Remember a vector field is what you get when you assign every single point in space to a vector As always I'm gonna use color because the lines get too long If you recall a line integral also deals with the curve, which I'll call C The curve has a few requirements though .First, it must be closed and it should be oriented counterclockwise I'm gonna call the region inside this curve R Green's theorem states that the line integral of F over C is equal to the double integral of the two-dimensional curl of F over R The curl of a vector field in two dimensions gives a sense of rotation over a region in the vector field The curl of a vector field can now describe how clockwise or counterclockwise the region is Positive curl indicates counterclockwise rotation and likewise negative curl indicates clockwise rotation Nonzero curl doesn't always occur in vector fields that look like circles For example, we have a region with higher magnitude vectors on the top and lower Magnitude vectors on the bottom this results in an overall clockwise rotation, which is negative curl We can represent curl as a cross product between the Del vector which by the way is every partial derivative and the vector field For now, let's not go over why but I highly suggest watching this video by 3blue1brown if you're curious First let's look at the line integral of a vector field Here's the important bit. If the overall rotation of the region R is counterclockwise.,the line integral will be positive and Negative if the overall rotation is clockwise Now let's look at our line integral What would happen if we split it into two curves? The resulting integral is the line integral of F over c1 Added to the line integral of F over c2 if we look at this, it's the same as the line integral over C! This is because the center line cancels out One is going from up to down and other one is going from down to up This is a really important fact. It means that we can split our curve into any amount of curves And as long as they add up to the same curve We can add the line integrals together and get the line integral of the outside curve Now let's split our region into many many small regions if you recall we have another way of describing rotation of a vector field - curl! As we described curl is for points in a vector field. So, we can approximate our line integral by summing up the curl times a small region for every single point on our region This is just a double integral! Let's go over that again. In calculus, we always use this idea of fusing infinite approximations For example, we did this one learning about the definite integral. Using Riemann sums the area under curve was approximated and the integral was the sum of these areas as we approached infinite number of rectangles In the beginning we established that we can split a curve into as many pieces as we want and some of the line integrals of each Of these pieces to give the line integral of the whole curve. In a calculus sense, We might see what happens when we reuse a ton of more rectangles. As the size of the rectangle Approaches zero, the curl of the vector field becomes a much much better approximation For the line integral of each of the small pieces Thus we can sum up the curl of every point inside the region of R to get the line integral of the whole curve If it still sounds confusing. I've left a few links in the description to articles that talk about this We see this kind of approximation into equality all over calculus It started with limits and how a function can approach a value. This is essentially the same thing Now we have the line integral over vector field described as the double integral of the curl of the same vector field and This should make complete sense. What's really interesting is that the double integral, Which is related to the region inside the curve can only be described that by looking at the edges! This is analogous to the fundamental theorem of calculus from single variable calculus If you remember the states that the area under a curve can just be described by looking at the endpoints Similarly, the region inside a curve can be described by just looking at its edges Now let's look at an example of how this can be used Let's say we have a vector field F = <6y-9x, -yx+x^3> And the curve C is the curve over here Let's try to calculate the line integral of F over C Green's theorem states that the line integral of this vector field over C is equal to the double integral of the 2-dimensional curl over the region inside of that. So first, let's calculate the two-dimensional curl of the vector field This is equal to x-9 Now to value at this line integral, we need to evaluate the double integral of x-9 over R the top line is y = 3-x and the bottom line is y = -1 The X limits are from X= -1 to 1 So the double integral is equal to this Plugging it in we get -218/3 The next videos are going to talk about this in three dimensions, which you might have heard as called Stokes theorem I'll also be talking about the divergence theorem. So stay tuned! Thank you for watching <3

Channel: vcubingx

Views: 267,654

Keywords: VCubing X, vcubingx, Green's Theorem, calculus, multivariable calculus, stokes', theorem, divergence, curl, calculus 3, derivative, integral, line integral, double integral, del, partial derivative, manim, 3b1b

Id: 8SwKD5_VL5o

Length: 6min 31sec (391 seconds)

Published: Mon Jun 10 2019