Find the gradient vector field of $f$.

$$f(x, y)=\ln (x+2 y)$$

Find the indicated partial derivatives. $f(x, y)=\ln \left(x^{2}+y^{2}\right)+2 ; \quad f_{x}(x, y), f_{x r}(x, y), f_{x y}(x, y)$

Find the gradients of the following functions:

$$f(x,y,z)=x^2+y^3+z^4$$

$$f(x,y,z)=x^2y^3z^4$$

$$f(x,y,z)=e^x\sin(y)\ln(z)$$

Use the gradient to find the directional derivative of the function at P in the direction of v. f(x, y, z) = x² + y² + z², P(1, 1, 1), v = √3/3(i - j + k)

Beam from a Lighthouse A light in a lighthouse 2000 m from a straight shoreline is rotating at 2 revolutions per minute. How fast is the beam moving along the shore when it passes a point 500 m from the point on the shore opposite the lighthouse?

A lighthouse stands 500 m off a straight shore and the focused beam of its light revolves four times each minute. As shown in the figure, P is the point on shore closest to the lighthouse and Q is a point on the shore 200 m from P. What is the speed of the beam along the shore when it strikes the point Q? Describe how the speed of the beam along the shore varies with the distance between P and Q. Neglect the height of the lighthouse.

Find the arc length function for the curve y=2x^3/2 with starting point P0(1,2).

Use a graph of the vector field $\mathbf{F}$ and the curve $C$ to guess whether the line integral of $\mathbf{F}$ over $C$ is positive, negative, or zero. Then evaluate the line integral.

$\mathbf{F}(x, y)=(x-y) \mathbf{i}+x y \mathbf{j}$, $C$ is the arc of the circle $x^2+y^2=4$ traversed counterclockwise from $(2,0)$ to $(0,-2)$

Prove the identity, assuming that the appropriate partial derivatives exist and are continuous. If $f$ is a scalar field and $\mathbf{F}, \mathbf{G}$ are vector fields, then $f \mathbf{F}, \mathbf{F} \cdot \mathbf{G}$, and $\mathbf{F} \times \mathbf{G}$ are defined by

$$\begin{aligned} (f \mathbf{F})(x, y, z) & =f(x, y, z) \mathbf{F}(x, y, z) \\ (\mathbf{F} \cdot \mathbf{G})(x, y, z) & =\mathbf{F}(x, y, z) \cdot \mathbf{G}(x, y, z) \\ (\mathbf{F} \times \mathbf{G})(x, y, z) & =\mathbf{F}(x, y, z) \times \mathbf{G}(x, y, z) \end{aligned}$$

$$\operatorname{div}(\mathbf{F}+\mathbf{G})=\operatorname{div} \mathbf{F}+\operatorname{div} \mathbf{G}$$

Evaluate the surface integral $\iint_S \mathbf{F} \cdot d \mathbf{S}$ for the given vector field $\mathbf{F}$ and the oriented surface $S$. In other words, find the flux of $\mathbf{F}$ across $S$. For closed surfaces, use the positive (outward) orientation.

$\mathbf{F}(x, y, z)=x z e^y \mathbf{i}-x z e^y \mathbf{j}+z \mathbf{k}$, $S$ is the part of the plane $x+y+z=1$ in the first octant and has downward orientation

Suppose $S$ and $C$ satisfy the hypotheses of Stokes' Theorem and $f, g$ have continuous second-order partial derivatives.

$$\int_C(f \nabla g+g \nabla f) \cdot d \mathbf{r}=0$$

Suppose $S$ and $C$ satisfy the hypotheses of Stokes' Theorem and $f, g$ have continuous second-order partial derivatives.

$$\int_c(f \nabla f) \cdot d \mathbf{r}=0$$

Suppose $S$ and $C$ satisfy the hypotheses of Stokes' Theorem and $f, g$ have continuous second-order partial derivatives.

$$\int_C(f \nabla g) \cdot d \mathbf{r}=\iint_S(\nabla f \times \nabla g) \cdot d \mathbf{S}$$

If a circle $C$ with radius 1 rolls along the outside of the circle $x^2+y^2=16$, a fixed point $P$ on $C$ traces out a curve called an epicycloid, with parametric equations $x=5 \cos t-\cos 5 t, y=5 \sin t-\sin 5 t$. Graph the epicycloid and use (5) to find the area it encloses.