Green’s, Stokes’, and the Divergence Theorems

Trivium

Given a vector field over either $\mathbf{R}^2$ or $\mathbf{R}^3,$ compute its divergence and its curl.

Compute the value of a line integral over a piecewise-smooth curve in a vector field. If the curve is closed, compute its value either directly or as an integral over its interior via either Green’s Theorem or Stokes’ Theorem.

Compute the value of a surface integral over a piecewise smooth surface in a vector field. If the surface is closed, compute its value either directly or as an integral over its interior via either Green’s Theorem or the Divergence Theorem.

Exercises

1. TK MORE HERE
2. For $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ compute the curl of $\bm{F}.$
3. Show that the vector field $\bm{F}$ defined as $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ is not conservative.
4. Show that the vector field $\bm{F}$ defined as $\bm{F}(x,y,z) = y^2 z^3\mathbf{i} + 2xyz^3 \mathbf{j} +3xy^2z^2\mathbf{k}$ is a conservative vector field, and find an example of a function $f$ such that $\bm{F} = \nabla f.$
5. For $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ compute the divergence of $\bm{F}.$
6. Show that the vector field $\bm{F}$ defined as $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ cannot be written as the curl of another vector field. I.e. there is no vector field $\bm{G}$ such that $\bm{F} = \operatorname{curl}\bm{G}.$

Problems & Challenges

1. Schey Let $\mathbf{i}$ and $\mathbf{j}$ and $\mathbf{k}$ be the unit basis vectors of the standard rectangular coordinate system on $\mathbf{R}^3,$ and let $\mathbf{e}_r$ and $\mathbf{e}_\theta$ and $\mathbf{e}_z$ be the unit basis vectors of the standard cylindrical coordinate system on $\mathbf{R}^3.$
   1. Show that $$\begin{align*} \mathbf{i} &= \mathbf{e}_r\cos(\theta) - \mathbf{e}_\theta\sin(\theta) \\\mathbf{j} &= \mathbf{e}_r\sin(\theta) + \mathbf{e}_\theta\cos(\theta) \\\mathbf{k} &= \mathbf{e}_z\,. \end{align*}$$
   2. For a vector field $\bm{F}(r,\theta,z) = P\mathbf{e}_r + Q\mathbf{e}_\theta + R\mathbf{e}_z,$ show that $\operatorname{div}\bm{F}$ is given by $$\operatorname{div}\bm{F} = \frac{1}{r}\frac{\partial}{\partial r}\big(rP\big) + \frac{1}{r}\frac{\partial Q}{\partial \theta} + \frac{1}{r}\frac{\partial R}{\partial z}\,.$$
   3. Determine an analogous formula for $\operatorname{curl}\bm{F}$ in cylindrical coordinates.
2. Schey Let $\mathbf{i}$ and $\mathbf{j}$ and $\mathbf{k}$ be the unit basis vectors of the standard rectangular coordinate system on $\mathbf{R}^3,$ and let $\mathbf{e}_\rho$ and $\mathbf{e}_\theta$ and $\mathbf{e}_\varphi$ be the unit basis vectors of the standard spherical coordinate system on $\mathbf{R}^3.$
   1. Show that $$\begin{align*} \mathbf{i} &= \mathbf{e}_\rho\sin(\theta)\cos(\varphi) + \mathbf{e}_\theta\cos(\theta)\cos(\varphi) - \mathbf{e}_\varphi\sin(\varphi) \\\mathbf{j} &= \mathbf{e}_\rho\sin(\theta)\sin(\varphi) + \mathbf{e}_\theta\cos(\theta)\sin(\varphi) - \mathbf{e}_\varphi\cos(\varphi) \\\mathbf{k} &= \mathbf{e}_\rho\cos(\theta) - \mathbf{e}_\theta\sin(\theta) \end{align*}$$
   2. For a vector field $\bm{F}(\rho,\theta,\varphi) = P\mathbf{e}_\rho + Q\mathbf{e}_\theta + R\mathbf{e}_\varphi,$ show that $\operatorname{div}\bm{F}$ is given by $$\operatorname{div}\bm{F} = \frac{1}{\rho^2}\frac{\partial}{\partial \rho}\big(r^2P\big) + \frac{1}{\rho\sin(\theta)}\frac{\partial}{\partial \theta}\big(\sin(\theta)Q\big) + \frac{1}{\rho\sin(\theta)}\frac{\partial R}{\partial \varphi}$$
   3. Determine an analogous formula for $\operatorname{curl}\bm{F}$ in spherical coordinates.
3. Schey For an arbitrary vector $\bm{v}$ and $\bm{r} = \langle x,y,z\rangle,$ show that $$\operatorname{curl}\Bigl(\frac{\bm{v}\times\bm{r}}{2}\Bigr) = \bm{v}\,.$$

4. Schey Verify Stokes’ Theorem for these surfaces $S$ and vector fields $\bm{F}.$

   1. The surface $S$ consisting of the five faces of the unit cube in the first quadrant that don’t lie in the $xz$ plane, and $\bm{F} = \langle z^2, -y^2, 0\rangle.$
   2. The surface that is the eighth of the unit sphere centered at the origin that lies entirely inside the first quadrant, and $\bm{F} = \langle y,z,x \rangle.$
5. Stewart Prove the following identity: $$\nabla\bigl(\bm{F}\cdot\bm{G}\bigr) = \bigl(\bm{F}\cdot\nabla\bigr) \bm{G} + \bigl(\bm{G}\cdot\nabla\bigr) \bm{F} + \bm{F} \times \operatorname{curl}\bm{G} + \bm{G} \times \operatorname{curl}\bm{F}$$

6. Schey Recalling that $\operatorname{div} = (\nabla\cdot)$ and that $\operatorname{grad} = (\nabla)$ and that $\operatorname{curl} = (\nabla\times)$ and that $\nabla^2$ is the Laplace operator, verify the following identities concerning arbitrary differentiable scalar functions $f$ and $g$ and arbitrary differentiable vector fields $\bm{F}$ and $\bm{G}.$

   $\nabla(fg) = f\nabla g + g\nabla f$

   $\nabla\cdot \bigl(f\bm{F}\bigr) = f\nabla\cdot\bm{F} + \bm{F}\cdot\nabla f$

   $\nabla\cdot\bigl(\bm{F}\times\bm{G}\bigr) = \bm{G} \cdot \bigl(\nabla\times\bm{F}\bigr) - \bm{F} \cdot \bigl(\nabla\times\bm{G}\bigr)$

   $\nabla\times \bigl(f\bm{F}\bigr) = f\nabla\times\bm{F} - \bm{F}\times\nabla f$

   $\nabla\times\bigl(\bm{F}\times\bm{G}\bigr) = \bigl(\bm{G}\cdot\nabla\bigr)\bm{F} - \bigl(\bm{F}\cdot\nabla\bigr)\bm{G} + \bm{F}\bigl(\nabla\cdot\bm{G}\bigr) - \bm{G}\bigl(\nabla\cdot\bm{F}\bigr)$

   $\nabla\times\bigl(\nabla\times\bm{F}\bigr) = \nabla\bigl(\nabla\cdot\bm{F}\bigr) - \nabla^2\bm{F}$

7. Stewart Find the positively oriented simple closed curve $C$ for which the value of the line integral is a maximum. $$\oint\limits_C \bigl(y^3-y\bigr)\,\mathrm{d}x - \bigl(2x^3\bigr)\,\mathrm{d}y$$