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.

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).

$$\int_{(-1,1)}^{(4,2)}\left(y-\frac{1}{x^2}\right) d x+\left(x-\frac{1}{y^2}\right) d y$$

Find the center of mass of the homogeneous triangle with vertices (a, 0,0),(0, b, 0), and (0,0, c), where a,b, and c are all positive.

If $\mathbf{F}=\left(x^2+y^2\right) \mathbf{i}+2 x y \mathbf{j}$, calculate the circulation of $\mathbf{F}$ around C of given problem; that is, calculate $\oint_C \mathbf{F} \cdot \mathbf{T} d s$.

In given problem, evaluate each line integral.

Same integral as in previous problem; C is the line segment from (0,0,0) to (2,3,4).

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

$$\mathbf{F}(x, y, z)=\cos x \mathbf{i}+\sin y \mathbf{j}+3 \mathbf{k}$$

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

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

Find the work done by $\mathbf{F}=\left(x^2+y^2\right) \mathbf{i}-2 x y \mathbf{j}$ in moving a body counterclockwise around the curve C of given problem.

Let C be the circle that is the intersection of the plane $a x+b y+z=0(a \geq 0, b \geq 0)$ and the sphere $x^2+y^2+z^2=9$. For $\mathbf{F}=y \mathbf{i}-x \mathbf{j}+3 y \mathbf{k}$, evaluate

$$\oint_C \mathbf{F} \cdot \mathbf{T} d s$$

Hint: Use Stokes's Theorem.

Let $\mathbf{F}=2 \mathbf{i}+x z \mathbf{j}+z^3 \mathbf{k}$ and $\partial S$ be the boundary of the surface $z=x y^2, 0 \leq x \leq 1,0 \leq y \leq 1$, oriented counterclockwise as viewed from above. Evaluate $\oint_{a S} \mathbf{F} \cdot \mathbf{T} d s$.

Find the center of mass of the homogeneous triangle with vertices (a, 0,0),(0, a, 0), and (0,0, a).

Let $\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}$, and let S be a solid for which Gauss's Divergence Theorem applies. Show that the volume of S is given by

$$V(S)=\frac{1}{3} \iint_{\lambda S} \mathbf{F} \cdot \mathbf{n} d S$$

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).

$$\int_{(2,1)}^{(6,3)} \frac{x^3}{\left(x^4+y^4\right)^2} d x+\frac{y^3}{\left(x^4+y^4\right)^2} d y$$

In given problem, evaluate each line integral.

$$\int_C(x+y+z) d x+(x-2 y+3 z) d y+$$

$(2 x+y-z) d z ; C$ is the line-segment path from $(0,0,0)$ to $(2,0,0)$ to $(2,3,0)$ to $(2,3,4)$.