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. Evaluate $\int_C x^4\,\mathrm{d}x + xy\,\mathrm{d}y,$ where $C$ is the triangular curve consisting of the line segments from $(0,0)$ to $(1,0)$ and from $(1,0)$ to $(0,1)$ and from $(0,1)$ to $(0,0).$
2. Evaluate $\oint \Big(3y-\mathrm{e}^{\sin(x)}\Big)\,\mathrm{d}x + \Big(7x+\sqrt{y^4+1}\Big)\,\mathrm{d}y,$ where $C$ is the circle $x^2+y^2=9.$
3. Calculate the area enclosed by the ellipse $\frac{1}{a^2}x^2+\frac{1}{b^2}y^2=1.$
4. Evaluate $\oint y^2\,\mathrm{d}x + 3xy\,\mathrm{d}y,$ where $C$ is the boundary of the semiannular region $R$ in the upper half-plane between the circles $x^2+y^2=1$ and $x^2+y^2=4.$
5. For the vector field $\bm{F}$ defined as

   $\displaystyle \bm{F}(x,y) = \frac{-y}{x^2+y^2}\mathbf{i}+\frac{x}{x^2+y^2}\mathbf{j}$

   show that $\int_C \bm{F}\cdot\mathrm{d}\bm{r} = 2\pi$ for every positively oriented simple closed path that encloses the origin.
6. For $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ compute the curl of $\bm{F}.$
7. 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.
8. 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 $\bm{F} = \nabla f.$
9. For $\bm{F}(x,y,z) = xz\mathbf{i} + xyz\mathbf{j} -y^2\mathbf{k}$ compute the divergence of $\bm{F}.$
10. 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 $\bm{F} = \operatorname{curl}\bm{G}.$
11. Evaluate $\int_C \bm{F}\cdot\mathrm{d}\bm{r},$ where $\bm{F}(x,y,z) = -y^2\mathbf{i} +x\mathbf{j}+z^2\mathbf{k}$ and $C$ is the curve of intersection of the plane $y+z=2$ and the cylinder $x^2+y^2=1.$ (Orient $C$ to be counterclockwise when viewed from above.)
12. Use Stokes’ Theorem to compute the integral $\iint_S \operatorname{curl}\bm{F}\cdot\mathrm{d}\bm{S},$ where $\bm{F}(x,y,z) = xz\mathbf{i}+yz\mathbf{j}+xy\mathbf{k}$ and $S$ is the part of the sphere $x^2+y^2+z^2=4$ that lies inside the cylinder $x^2+y^2=1$ and above the $xy$-plane.
13. Calculate the flux of the vector field $\bm{F}(x,y,z) = z\mathbf{i}+y\mathbf{j}+x\mathbf{k}$ across the unit sphere $x^2+y^2+z^2=1.$
14. Evaluate $\iint_S \bm{F}\cdot\mathrm{d}\bm{S},$ where $$\bm{F}(x,y,z) = xy\mathbf{i} + \Big(y^2+\mathrm{e}^{xz^2}\Big)\mathbf{j} + \sin(xy)\mathbf{k}$$ and $S$ is the surface of the region $E$ bounded by the parabolic cylinder $z=1-x^2$ and the planes $z=0$ and $y=0$ and $y+z=2.$

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. $$\oint\limits_C \bigl(y^3-y\bigr)\,\mathrm{d}x - \bigl(2x^3\bigr)\,\mathrm{d}y$$