Task 44.1

What do the curl and divergence of a vector field compute? How do we physically interpret them?

  1. Head to the following webpages and read their descriptions.
  2. After reading the 4 pages above, head to Wikipedia's article on magnetic fields, see https://en.wikipedia.org/wiki/Magnetic_field. Skim through the article (you do not need to read and comprehend all of it).
    • Are there any new symbols you don't recognize? What are they?
    • Search for the word "divergence" in the text, and read the paragraph(s) where you find it.
    • Search for the word "curl" in the text, and read the paragraph(s) where you find it.
    • Come with questions about anything you would like to discuss.

Task 44.2

As we proceed through the remainder of this unit, we'll find that three types of integrals appear quite often.

  • Work integrals: $\ds\int_{C} Mdx+Ndy+Pdz$
  • Surface integrals: $\ds \iint_S f(x,y,z) dS$
  • Flux integrals: $\ds \iint_S \vec F \cdot \hat n dS$

For each type of integral above, use Mathematica to create a code chunk that you can use to compute each type of integral. This will allow you free yourself from performing tedious computations, letting you focus on bigger picture ideas.

  • For the work integral, you can pick the vector field $\vec F(x,y,z)$, parametrization $\vec r(t)$, and bounds for $t$. Feel free to use an example from 42.3.
  • For the surface integral, you can pick the surface $S$, function $f(x,y,z)$, parametriztion $\vec r(u,v)$, and bounds for $u$ and $v$. Feel free to use an example from 42.2.
  • For the flux integral, you can pick the surface $S$, orientation $\hat n$, parametriztion $\vec r(u,v)$, and bounds for $u$ and $v$. Feel free to use an example from 42.1 or 42.3.

The ReplaceAll[] command in Mathematica will allow you to replace any instance of $x$, $y$, or $z$ with the corresponding value from your parametrization. This allows you to enter the vector field using Cartesian coordinates. Other commands you'll find useful are D[], Cross[], Norm[], Dot[], and Integrate[].

As an example, to dot the vector field $\vec F = (3x+4yz, 2y^2, x-4z)$ with the derivative of the parametrization $\vec r(t) = (3\cos t, 3\sin t, 4 t)$, and then make sure everything is in terms of $t$, we can use the following code. 

F = {3 x + 4 y z, 2 y^2, x - 4 z}
r = {3 Cos[t], 3 Sin[t], 4 t}
ReplaceAll[Dot[F, D[r, t]], {x -> r[[1]], y -> r[[2]], z -> r[[3]]}]

(*Or using short cut notation (. for Dot and /. for ReplaceAll)*)
F . D[r, t] /. {x -> r[[1]], y -> r[[2]], z -> r[[3]]}

The notation r[[1]] accesses the first element from r, which is 3 Cos[t].

Task 44.3

Green's theorem connects circulation along a simple closed curve in 2D to an integral along the region inside the curve. The circulation per area (circulation density) is given by $N_x-M_y$ for a 2D vector field. A similar fact is true in 3D, namely if $S$ is a surface with a simple closed curve $C$ that forms the boundary of $S$, then the circulation along $C$ can be computed instead by summing circulation densities (circulation per surface area) along the surface. Given a point $(x,y,z)$ and a unit vector $\hat n$, we calculate the circulation density of $\vec F$ about $\hat n$ at $(x,y,z)$ in a similar manner, as follows.

  • We construct a circle $C_a$ of radius $a$ in the plane through $(x,y,z)$ with normal vector $\hat n$.
  • We compute the counter-clockwise circulation about $\hat n$, obtaining $\oint_{C_a} Mdx+Ndy+Pdz$.
  • We divide by the area $A_a$ inside $C_a$, to obtain a circulation per area.
  • The circulation density of $\vec F$ about $\hat n$ at $(x,y,z)$ is the limit $$\lim_{a\to 0}\frac{1}{A_a}\oint_{C_a} Mdx+Ndy+Pdz = \vec \nabla \times \vec F \cdot \hat n.$$

The curl of a vector field provides information about circulation density in every direction (just dot by the direction about which you want to know how things are rotating). Note that for a 2D vector field, we can use the normal vector $\hat n = (0,0,1)$, and then we see that the circulation density for a vector field in 2D is $$(N_z-P_y, P_x-M_z, N_x-M_y) \cdot (0,0,1) = N_x-M_y.$$

We're now ready to state Stokes's theorem.

Stokes's Theorem

Given an orientable differentiable parametric surface with unit normal vector (orientation) $\hat n$ and piecewise smooth boundary $\partial S$, where each portion of the boundary is oriented compatibly with $\hat n$ (see below for meaning), then the flux of the curl of $\vec F=(M,N,P)$ is equal to the circulation of $\vec F$ along $\partial S$, which we can write as $$\iint_S \vec \nabla \times \vec F \cdot \hat n dS = \int_{\partial S} Mdx+Ndy+Pdz.$$

  • We use $\partial S$ to stand for the boundary of $S$. We are not taking partial derivatives of the surface, rather it's a notation convention.
  • Note that $\partial S$ might consist of multiple curves, which may or may not be connected to each other. Sum $\int_{C} Mdx+Ndy+Pdz$ for each curve $C$ that is part of the boundary to compute $\int_{\partial S} Mdx+Ndy+Pdz$.
  • We say that a curve $C$ is oriented compatibly with $\hat n$ provided that the surface is on the left side of $C$ when viewing the surface from the side designated by $\hat n$. As you wrap your right hand about the vector $\hat n$, your hand should be moving in the same direction as the orientation on $C$.

Let's verify this theorem in some examples.

  1. Let $\vec F(x,y,z) = (x+2y, 3x-4z, 2z)$. Consider the surface $S$ parametrized by $\vec r(u,v) = (u\cos v, u\sin v, u^2)$ for $0\leq u\leq 3$ and $0\leq v\leq 2\pi$ with orientation $\hat n$ pointing away from the $z$-axis. The boundary $\partial S$ consists of a single curve $C_1$ parametrized by $\vec r_1(t) = (3\cos t, 3\sin t, 9)$ for $0\leq t\leq 2\pi$.
    1. Draw the surface and curve. Do your best to explain what it means for $\hat n$ and the curve to be oriented compatibly.
    2. Set up and compute the surface integral $\ds \iint_S \vec \nabla \times \vec F \cdot \hat n dS$.
    3. Set up and compute the line integral $\ds \int_{\partial S} Mdx+Ndy+Pdz$. Do you get the same result as the surface integral?
  2. Let $\vec F(x,y,z) = (x+2y, 2x+4z, 4y)$. Consider the plane in the first octant with the three vertices $(3,0,0)$, $(0,3,0)$ and $(0,0,3)$. An equation of the plane is $x+y+z=3$, which we can parameterize using $\vec r(u,v) = (u,v,3-u-v)$ for $0\leq u\leq 3$ and $0\leq v\leq 3-u$. The boundary $\partial S$ consists of 3 curves, which we can parameterize using $$\begin{align*} \vec r_1(t) &= (3-3t,3t,0) \quad \text{for} 0\leq t\leq 1,\\ \vec r_2(t) &= (0,3-3t,3t) \quad \text{for} 0\leq t\leq 1,\\ \vec r_3(t) &= (3t,0,3-3t) \quad \text{for} 0\leq t\leq 1. \end{align*}$$
    1. Draw the surface and 3 curves, placing an arrow on the curves to demonstrate the direction of motion.
    2. Compute $\ds \int_{\partial S} Mdx+Ndy+Pdz$ by computing the three line integrals and summing the result.
    3. Compute $\ds \iint_S \vec \nabla \times \vec F \cdot \hat n dS$, using an orientation $\hat n$ that is compatible with the orientation of the curves.

Task 44.4

Pick some problems related to the topics we are discussing from the Text Book Practice page.