Are the statements true or false? Give reasons for your answer.

If a vector field $\vec{F}$ in $3$-space has zero divergence then $\vec{F} = a\vec{i} + b\vec{j} + c\vec{k}$ where $a, b$ and $c$ are constants.

Evaluate both sides of the divergence theorem for the vector field $\mathbf{H}=2 x y \mathbf{a}_{x}+\left(x^{2}+z^{2}\right) \mathbf{a}_{y}+2 y z \mathbf{a}_{z}$ and the rectangular region defined by $0<x<1,1<y<2,-1<z<3$

Verify that the Divergence Theorem is true for the vector field $\mathbf{F}$ on the region $E$. $\mathbf{F}(x, y, z)=x^{2} \mathbf{i}+x y \mathbf{j}+z \mathbf{k}$, E is the solid bounded by the paraboloid $z=4-x^{2}-y^{2}$ and the $x y-$plane

Consider the integral $\int_0^3 \frac{10}{x^2-2 x} d x$.
To determine the convergence or divergence of the integral, how many improper integrals must be analyzed? What must be true of each of these integrals if the given integral converges?

Prove that the divergence of a curl is always zero.

In this exercise, determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=1}^{\infty} \frac{1 \cdot 3 \cdot 5 \cdot \cdots(2 n-1)}{2 \cdot 5 \cdot 8 \cdots(3 n-1)}$$

In this exercise, the terms of a series $\sum_{n=1}^{\infty} a_n$ are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning.

$$a_1=\dfrac{1}{5}, a_{n+1}=\dfrac{\cos n+1}{n} a_n$$

In this exercise, the terms of a series $\sum_{n=1}^{\infty} a_n$ are defined recursively. Determine the convergence or divergence of the series. Explain your reasoning.

$$a_1=1, a_{n+1}=\dfrac{\sin n+1}{\sqrt{n}} a_n$$

Assume that S and T satisfy the conditions of the Divergence Theorem.

Show that the volume of T is given by V(T)=1/3 of the surface integral of the vector r dot product n dS, where r= xi + yj + zk.

Evaluate $\iint_{\partial} \mathbf{F} \cdot \mathbf{n} d A$, where $\mathbf{F}(x, y, z)=x \mathbf{i}+$ $y \mathbf{j}-z \mathbf{k}$ and $W$ is the unit cube in the first octant. Perform the calculation directly and check by using the divergence theorem.

Verify that the Divergence Theorem is true for the vector field $\mathbf{F}$ on the region $E$.

$\mathbf{F}(x, y, z)=x^2 \mathbf{i}+x y \mathbf{j}+z \mathbf{k}$, $E$ is the solid bounded by the paraboloid $z=4-x^2-y^2$ and the $x y$-plane

Verify the divergence theorem on the unit cube $0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1$ for the given $\mathbf{v}$.

$$\mathbf{v}(x, y, z)=x^2 \mathbf{i}-x z \mathbf{j}+z^2 \mathbf{k}$$

Prove the following properties of the divergence and curl. Assume F and G are differentiable vector fields and $c$ is a real number. $\nabla \times(\mathbf{F}+\mathbf{G})=(\nabla \times \mathbf{F})+(\nabla \times \mathbf{G})$

Evaluate $\iint_{\partial W} \mathbf{F} \cdot \mathbf{n} d A,$ where $\mathbf{F}(x, y, z)=x \mathbf{i}+y \mathbf{j}-z \mathbf{k}$ and W is the unit cube in the first octant. Perform the calculation directly and check by using the divergence theorem.