$\vec{F} = (e^z + x)\vec{i} + (2y + \sin x)\vec{j} +\Big (e^{x^2}− z \Big)\vec{k}$ and $S$ is the sphere of radius $1$ centered at $(2, 1, 0)$.

Find the divergence of the vector field. (Note: $\vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$.)

$$\vec{F} (\vec{r} ) = \vec{a} \times \vec{r}$$

Let div $\vec{F} = 2(6 − x)$ and $0 \leq a, b, c \leq 10$.

Find the flux of $\vec{F}$ out of the rectangular solid $0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq c$.

Suppose that $\vec{A}$ is a vector potential for $\vec{B}$.

Show that $\vec{A} +$ grad $\psi$ is also a vector potential for $\vec{B}$ , for any function $\psi$ with continuous secondorder partial derivatives. (The vector potentials $\vec{A}$ and $\vec{A} +$ grad $\psi$ are called gauge equivalent and the transformation, for any $\psi$, from $\vec{A}$ to $\vec{A} +$ grad $\psi$ is called a gauge transformation.)

The divergence of a magnetic vector field $\vec{B}$ must be zero everywhere. Which of the following vector fields cannot be a magnetic vector field?

$$\vec{B} (x, y, z)=(x^2 − y^2 − x)\vec{i} + (y − 2xy)\vec{j}$$

The divergence of a magnetic vector field $\vec{B}$ must be zero everywhere. Which of the following vector fields cannot be a magnetic vector field?

$$\vec{B} (x, y, z) = −z\vec{i} + y\vec{j} + x\vec{k}$$

$\vec{F} = x^2\vec{i} + y^2\vec{j} + e^{xy}\vec{k}$ and $S$ is the cube of side $3$ in the first octant with one corner at the origin, and sides parallel to the axes.

Evaluate the line integral, where C is the given curve. $\int_{C} \sin x d x+\cos y d y, \quad C$ consists of the top half of the circle $x^{2}+y^{2}=1$ from (1, 0) to (-1, 0) and the line segment from (-1, 0) to (-2, 3)

Compute the line integral of the vector field over the oriented curve.

$\mathbf{F}=\langle y+z, x, y\rangle, \quad \mathbf{c}(t)=\left(t+t^2, \dfrac{1}{3} t^3, 2+t\right)$ for $0 \leq t \leq 2$

Compute the line integral of the scalar function over the curve.

f(x, y, z)=3zy-1+12xz, $\mathbf{c}(t)=\left(t, \dfrac{t^2}{\sqrt{2}}, \dfrac{t^3}{3}\right), \quad 0 \leq t \leq 2$

Compute the line integral of the vector field over the oriented curve.

$\mathbf{F}=\left\langle x^2, x y\right\rangle, \quad$ line segment from $(0,0)$ to $(2,2)$

Evaluate the line integral, where C is the given curve. integral C xy^4 ds, C is the right half of the circle x^2+y^2=16

The divergence of a magnetic vector field $\vec{B}$ must be zero everywhere. Which of the following vector fields cannot be a magnetic vector field?

$$\vec{B} (x, y, z) = −y\vec{i} + x\vec{j} + (x + y)\vec{k}$$

The magnetic field, $\vec{B}$ , due to a magnetic dipole with moment $\vec{\mu}$ satisfies div $\vec{B} = 0$ and is given by

$\vec{B} = − \frac{\vec{\mu}}{||\vec{r}||^3} + \frac{3(\vec{\mu} \cdot \vec{r} )\vec{r}}{||\vec{r}||^5}, \qquad \vec{r} \ne \vec{0}$.

Does your answer to part $(a)$ contradict your answer to part $(b)$? Explain.