Use Green’s first identity to show that if $f$ is harmonic on $D,$ and if $f(x, y) = 0$ on the boundary curve $C,$ then $\iint_D\lvert \triangledown f\rvert^2~dA = 0$. (Assume the same hypotheses as in the previous exercise.)

Sketch the vector field $\mathbf{F}$ by drawing a diagram like the earlier figures.

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

Let $\text{F}(x, y) = \dfrac{-y\text{i} + x\text{j}}{x^2 + y^2}$. Show that $\partial P/\partial y = \partial Q/\partial x$.

Find the mass and center of mass of a wire in the shape of the helix $x = t, y = \cos t, z = \sin t, 0 \leqslant t \leqslant 2\pi,$ if the density at any point is equal to the square of the distance from the origin.

At time $t = 1,$ a particle is located at position (1, 3). If it moves in a velocity field

$$\text{F}(x, y) = \lang xy - 2, y^2 - 10\rang$$

find its approximate location at time $t = 1.05$.

Find a formula for $\iint_S \mathrm{F} \cdot d \mathrm{S}$ similar to Formula 10 for the case where $S$ is given by $x=k(y, z)$ and $\mathrm{n}$ is the unit normal that points forward (that is, toward the viewer when the axes are drawn in the usual way).

Find the center of mass of a wire in the shape of the helix $x = 2\sin t, y = 2\cos t, z = 3t, 0 \leqslant t \leqslant 2\pi,$ if the density is a constant $k$.

Write the formulas similar to the appropriate equations for the center of mass $(\overline{x}, \overline{y}, \overline{z})$ of a thin wire in the shape of a space curve $C$ if the wire has density function $\rho(x, y, z)$.

Let

$$\mathrm{F}(x, y)=\frac{\left(2 x^3+2 x y^2-2 y\right) \mathrm{i}+\left(2 y^3+2 x^2 y+2 x\right) \mathrm{j}}{x^2+y^2}$$

Evaluate $\oint_C \mathrm{F} \cdot d \mathrm{r}$, where $C$ is shown in the figure.

Recall from the previous section that a function $g$ is called harmonic on $D$ if it satisfies Laplace’s equation, that is, $\triangledown^2g = 0$ on $D$. Use Green’s first identity (with the same hypotheses as in the previous exercise) to show that if $g$ is harmonic on $D,$ then $\oint_CD_\text{n}g~ds = 0$. Here $D_\text{n}g$ is the normal derivative of $g$ defined in the previous exercise.

Evaluate the line integral by two methods: using Green's Theorem.

$\displaystyle\oint_C x y d x+x^2 y^3 d y, C$ is the triangle with vertices $(0,0)$, $(1,0)$, and $(1,2)$

Determine whether or not the given set $\{(x, y)~|~(x y) \not = (2, 3)\}$ is (a) open, (b) connected, and (c) simply-connected.

A particle moves in a velocity field $\text{V}(x, y) = \lang x^2, x + y^2\rang$. If it is at position (2, 1) at time $t = 3,$ estimate its location at time $t = 3.01$.

Find a formula for $\iint_S \mathrm{F} \cdot d \mathrm{S}$ similar to earlier formula for the case where $S$ is given by $y=h(x, z)$ and $\mathrm{n}$ is the unit normal that points toward the left.