Let

$$\mathbf{F}=\frac{y}{x^2+y^2} \mathbf{i}-\frac{x}{x^2+y^2} \mathbf{j}=M \mathbf{i}+N \mathbf{j}$$

(a) Show that $\partial N / \partial x=\partial M / \partial y$.

(b) Show, by using the parametrization $x=\cos t, y=\sin t$, that $\oint_C M d x+N d y=-2 \pi$, where C is the unit circle.

(c) Why doesn't this contradict Green's Theorem?

Use Green's Theorem to prove the plane case of Theorem 14.3D; that is, show that $\partial N / \partial x=\partial M / \partial y$ implies that $\oint_C M d x+N d y=0$, which implies that $\mathbf{F}=M \mathbf{i}+N \mathbf{j}$ is conservative.

In given problem, show that the given line integral is independent of path (use Theorem $\mathrm{C}$ ) and then evaluate the integral (either by choosing a convenient path or, if you prefer, by finding a potential function f and applying Theorem A).

$$\int_{(0,1,0)}^{(1,1,1)}(y z+1) d x+(x z+1) d y+(x y+1) d z$$

A central force is one of the form $\mathbf{F}=f(\|\mathbf{r}\|) \mathbf{r}$, where f has a continuous derivative (except possibly at $\|\mathbf{r}\|=0$ ). Show that the work done by such a force in moving an object around a closed path that misses the origin is 0.

In given problem, plot the parametric surface over the indicated domain.

$\mathbf{r}(u, v)=2 u \mathbf{i}+3 v \mathbf{j}+\left(u^2+v^2\right) \mathbf{k} ;-1 \leq u \leq 1$, $-2 \leq v \leq 2$

A wire of constant density has the shape of the helix $x=a \cos t, y=a \sin t, z=b t, 0 \leq t \leq 3 \pi$. Find its mass and center of mass.

In given problem, find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=(y+z) \mathbf{i}+(x+z) \mathbf{j}+(x+y) \mathbf{k}$$

Let $\mathbf{F}$ be a constant vector field. Show that

$$\iint_{\lambda S} \mathbf{F} \cdot \mathbf{n} d S=0$$

for any "nice" solid S. What should we mean by "nice"?

Let $\mathbf{F}=2 z \mathbf{i}+2 y \mathbf{k}$, and let $\partial S$ be the intersection of the cylinder $x^2+y^2=a y$ with the hemisphere $z=\sqrt{a^2-x^2-y^2}$, $a>0$. Assuming distances in meters and force in newtons, find the work done by the force $\mathbf{F}$ in moving an object around $\partial S$ in the counterclockwise direction as viewed from above.

Show that the work done by a constant force $\mathbf{F}$ in moving a body around a simple closed curve is 0.

Consider the plane ax+by+cz=d, where a, b, c, and d are all positive. Use previous problem to show that the volume of the tetrahedron cut from the first octant by this plane is $d D /\left(3 \sqrt{a^2+b^2+c^2}\right)$, where D is the area of that part of the plane in the first octant.

In given problem, plot the parametric surface over the indicated domain.

$\mathbf{r}(u, v)=u \mathbf{i}+3 v \mathbf{j}+\left(4-u^2-v^2\right) \mathbf{k} ; 0 \leq u \leq 2$, $0 \leq v \leq 1$

In given problem, find div $\mathbf{F}$ and curl $\mathbf{F}$.

$$\mathbf{F}(x, y, z)=e^x \cos y \mathbf{i}+e^x \sin y \mathbf{j}+z \mathbf{k}$$

Find the mass of a wire with the shape of the curve $y=x^2$ between (-2,4) and (2,4) if the density is given by $\delta(x, y)=$ k|x|.