Determine whether the vector function is a gradient and, if so, find all functions $f$ with that gradient.

$$\left(y^2 z^3+1\right) \mathbf{i}+\left(2 x y z^3+y\right) \mathbf{j}+\left(3 x y^2 z^2+1\right) \mathbf{k}$$

Show that the rectangular box of maximum volume that can be inscribed in the sphere $x^2+y^2+z^2=a^2$ is a cube.

Use a differential to estimate the change in $T=$ $x^2 \cos \pi z-y^2 \sin \pi z$ from $x=2, y=2, z=2$ to $x=2.1$, $y=1.9, z=2.2$.

Suppose that $f$ is differentiable at $\left(x_0, y_0\right)$ and $\nabla f\left(x_0, y_0\right) \neq$ $\mathbf{0}$. Calculate the rate of change of $f$ in the direction of

$$\frac{\partial f}{\partial y}\left(x_0, y_0\right) \mathbf{i}-\frac{\partial f}{\partial x}\left(x_0, y_0\right) \mathbf{j} \text {. }$$

Give a geometric interpretation to your answer.

Determine whether the vector function is a gradient and, if so, find all functions $f$ with that gradient.

$$(2 x \sin 2 y \cos z) \mathbf{i}+\left(2 x^2 \cos 2 y \cos z\right) \mathbf{j}-\left(x^2 \sin 2 y \sin z\right) \mathbf{k}$$

An object is moving along the curve in which the paraboloid $z=x^2+\frac{1}{4} y^2$ intersects the circular cylinder $x^2+y^2=13$. If the $x$-coordinate is increasing at the rate of 5 centimeters per second, how fast is the $z$-coordinate changing at the instant $x=2$ centimeters and $y=3$ centimeters?