35. The Vector Calculus Theorems

The fundamental theorem of calculus writes the value of the integral in terms of an antiderivative on the boundary of the integral.

$$\large \int_a^b f(x) dx=F(b)-F(a)$$

There are three big vector calculus theorems that follow this same pattern. In each case, we can rewrite an integral on a larger set in terms of an antidervative on a boundary of the set.

These theorems are essential for deriving the results of fluid dynamics and electromagnetics.

In our HW and test problems, we will use them to turn hard integrals into an easier ones.

Stokes Theorem

Stokes theorem lets us relate a flux integral to a line integral in the following way:

$$\large \oint_C \vec{F}\cdot d\vec{s} = \iint_S\left( \vec{\nabla}\times\vec{F}\right)\cdot d\vec{S}$$

Here, the curve $\large C$ is the boundary of the surface $\large S$. The picture below illustrates how we choose the orientation of the curve. Stand in the place of the normal vector and walk around $\large C$ in the direction that keeps the surface on your left.

^{Image from: https://mccarthymat501.commons.gc.cuny.edu/calculus-i-iii/mat-303-calculus-iii/mat-303-stokes-theorem/}

As an illustration, here are a few problems where we would use Stokes theorem.

1. Let $\large \vec{F}=\left<-y,2x,x+z\right>$. Find $\large \iint_S \vec{\nabla}\times\vec{F}\cdot d\vec{S}$ if $\large S$ is the upper hemisphere of radius 1. In this case, rather than dealing with the curl of $\large \vec{F}$, the surface parameterization, etc, we can use Stokes theorem to write this as a line integral around a circle of radius 1. This second option is easier!

2. Find $\large \oint_C \vec{F}\cdot d\vec{s}$ with $\large \vec{F}=\left<\sin(x^2),e^{y^2}+x,z^4+2x^2\right>$ if $\large C$ is the boundary of the part of the plane $\large 2x+3y+6z=6$ in the first octant. This boundary consists of three lines, so we will really need to do three line integrals! Also, the vector field looks awful to work with. But the curl of this vector field is nice, $\large \vec{\nabla}\times\vec{F}=\left<0,4x,2x\right>$, so it will be much easier for us to use Stokes theorem to write this as a flux integral of the curl of $\large \vec{F}$.

Green's Theorem

The most common implementation of Stokes theorem is for a 2D vector field. In this situation, we call the theorem Green's theorem instead. (Green was a math prodigy with a fascinating story: https://en.wikipedia.org/wiki/George_Green_(mathematician)). Green's theorem is:

$$ \displaystyle \large \oint_C \vec{F}\cdot d\vec{s} = \iint_R \left(\frac{\partial F_2}{\partial x}-\frac{\partial F_1}{\partial y}\right)dA $$

Here, the line integral is for a closed curve in the $\large xy$-plane. The surface integral in this 2D scenario to a double integral on the $\large xy$-plane. The domain $\large R$ is the region encolsed by the curve $\large C$. The combination of derivatives in the double integral is just the 2D curl (or, it's the $\large z$-component of the curl of two 2D vectors).

Like Stokes theorem, we will use Green's theorem to make integrals easier. Sometimes, we have a nasty line integral, and it turns out to be easier to do a double integral instead. That is the case in the following example:

In other examples, it may be easier to take the double integral of the curl of $\large \vec{F}$ and to write it as a line integral instead.

The Divergence Theorem

The final theorem relates a information about the divergence of the divergence of $\large \vec{F}$ on a 3D solid to information about $\large \vec{F}$ on the surface of the solid. It is the divergence theorem:

$$\large \iint_{\partial D}\vec{F}\cdot d\vec{S} = \iiint_D \vec{\nabla}\cdot \vec{F} dV$$

This is a triple integral on the right over some region $\large D$. The notation for the flux integral on the left, $\large \partial D$, indicates the surface of $\large W$.

This video illustrates a situation where finding the flux would require three surface integrals (with three surface parameterizations, three normal vectors, etc.) but the divergence theorem allows us to evaluate just one triple integral instead:

The End of Calculus!

And with that, congratulations, you now know all of the calculus! Of course, there is always more to learn, and we can always learn it deeper, but you should feel proud of all the calc you've mastered this year!

Tomorrow's class:

• Practice evaluating integrals using the three vector calc theorems.
• Justification for Green's theorem (view it as averages of vortices).

Section Outcomes:

I can:

• Use Stokes, Greens, and the divergence theorems to convert difficult integrals into easier integrals.

Other resources

For more on Green's theorem, see https://openstax.org/books/calculus-volume-3/pages/6-4-greens-theorem

For more on Stokes theorem, see https://openstax.org/books/calculus-volume-3/pages/6-7-stokes-theorem

For more on the divergence theorem, see https://openstax.org/books/calculus-volume-3/pages/6-8-the-divergence-theorem