Original and interesting problems about the theorems of Green, Stokes and Gauss

Question

Problems about these three classical theorems that we find in Calculus textbooks are usually in a low level. So, in this post, I'd like examples of original and interesting problems involving such theorems that can be solved with a typical Calculus 3 course background.

Answer 1

Probably not that hard but I found it interesting.

Prove the shoelace formula:

Let $(x_i,y_i)$ with $i=1,2,3,\cdots n$ be points in $\mathbb{R^2}$. Then the area of the simple polygon with such vertices is,

$$\frac{1}{2}\left(\sum_{i=1}^{n-1} x_iy_{i+1}+x_ny_1+\sum_{i=1}^{n-1} x_{i+1}y_i-x_1y_n \right)$$

Answer 2

Computing the area $A$ of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is not straightforward with usual techniques, but turns out to be very easy with Green's theorem:

On one hand $$ A = \iint_D\; dA $$ And on the other hand, Green's theorem states that $$ \oint_C P \;dx + Q\; dy = \iint \frac{\partial Q }{\partial x}-\frac{\partial P}{\partial y}\; dA $$

If we can find $P$ and $Q$ such that $\frac{\partial Q }{\partial x}-\frac{\partial P}{\partial y}=1$, we end up with $$ \oint_C P \;dx + Q\; dy = \iint \frac{\partial Q }{\partial x}-\frac{\partial P}{\partial y}\; dA = \iint_D\; dA = A $$ For example, $Q=x/2$ and $P=-y/2$ are suitable, and the equation becomes $$ A = \oint_C \frac{x}{2}\; dy - \frac{y}{2}\; dx = \oint_t \frac{x(t)}{2}\; y'(t)dt - \frac{y(t)}{2}\;x'(t) dt $$ For the ellipse, $C$ can be parametrized with $x=a\cos t$ and $y=b \sin t$, $t\in [0,2\pi]$ and it is a simple exercise to compute the above integral, which gives $$ A=\pi ab $$

Answer 3

I consider amazing the planimeter (Here too). How it works implies directly the Green Theorem.