Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.)

$$w=3 x^2 y-y^3+y^2, \quad C: 25 x^2+y^2=25$$

In the case of independence integrate from $(0,0,0)$ to $(a, b, c)$. (Show the details of your work.)

$$3(x+y)^2(d x+2 d y)+d z$$

Water flows steadily into and out of a tank that sits frictionless wheels as shown in Fig. Determine the diameter $D$ so that the tank remains motionless if $F=0$.

Evaluate $\int_C f(\mathbf{r}) d s$ with $f$ and $C$ as follows. (Show the details.) $f=\sqrt{2+x^2+3 y^2}, \quad C: \mathbf{r}=\left[t, t, t^2\right], 0 \leqq t \leqq 3$.

In following problem, with $\mathbf{F}$ and $C$ as given, evaluate $\int_C \mathbf{F}(\mathbf{r}) \cdot d \mathbf{r}$ by the method that seems most suitable (direct integration, use of exactness or Green's theorem or Stokes's theorem). Recall that if $\mathbf{F}$ is a force, the integral gives the work done in a displacement. (Show the details of your work.)

$$\mathbf{F}=\left[\begin{array}{ll}x^2, & -2 y^2\end{array}\right], \quad C$$

the straight-line segment from $(4,2)$ to $(-3,5)$

Surface Integrals $\iint_S G(\mathbf{r}) d A$. Using $\iint_S G(\mathbf{r}) d A=\iint_R G(\mathbf{r}(u, v))|\mathbf{N}(u, v)| d u d v$ or $\iint_S G(\mathbf{r}) d A=\iint_{R^*} G(x, y, f(x, y)) \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2} d x d y$, evaluate thise integral for the given data. (Show the details.)

$$G=\left(x^2+y^2\right)^2-z^2, \quad S: \mathbf{r}=[u \cos v, \quad u \sin v, \quad 2 u], \quad 0 \leqq u \leqq 1,-\pi \leqq v \leqq \pi$$

Water flows steadily from a tank mounted on a cart. After the water jet leaves the nozzle of the tank, it falls and strikes a vane attached to another cart. The cart's wheels are frictionless, and the fluid is inviscid. (a) Determine the speed of the water leaving the tank, $V_1$, and the water speed leaving the cart, $V_2$. (b) Determine the tension in rope A. (c) Determine the tension in rope $B$. The nozzle area is 0.01 m$^2$ and the height of the tank is 6 ft.

Find the coordinates $\bar{x}, \bar{y}$ of the center of gravity of a mass of density $f(x, y)=1$ in a region $R$, where $R$ is the region $x^2+y^2 \leqq a^2$ in the first quadrant. Check the result using polar coordinates.

Evaluate $\int_C f(\mathbf{r}) d s$ with $f$ and $C$ as follows. (Show the details.) $f=x^2+y^2+z^2, \quad C:[\cos t, \sin t, 2 t], 0 \leqq t \leqq 4 \pi$.

Find a parametric representation of the following surface. (The answer gives one such representation and there are many others.) Find a normal vector. (Show the details.) Ellipsoid $x^2+y^2+\frac{1}{4} z^2=1$.

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.) $w=\cosh x, \quad R$ the triangle with vertices $(0,0),(4,2),(0,2)$

Surface Integrals $\iint_S G(\mathbf{r}) d A$. Using $\iint_S G(\mathbf{r}) d A=\iint_R G(\mathbf{r}(u, v))|\mathbf{N}(u, v)| d u d v$ or $\iint_S G(\mathbf{r}) d A=\iint_{R^*} G(x, y, f(x, y)) \sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2} d x d y$, evaluate thise integral for the given data. (Show the details.)

$$G=(1+9 x z)^{3 / 2}, \quad S: \mathbf{r}=\left[\begin{array}{lll}u, & v, & u^3\end{array}\right], \quad 0 \leqq u \leqq 1,-2 \leqq v \leqq 2$$

In the case of independence integrate from $(0,0,0)$ to $(a, b, c)$. (Show the details of your work.)

$$y e^{2 z} d y-z e^y d z$$

Surface Integral $\int_S \mathbf{F} \cdot \mathbf{n} d A$. Evaluate this integral by the divergence theorem for the following data. (Show the details. More such problems in the next problem set.)

$$\mathbf{F}=\left[\begin{array}{lll}\cos y, & \sin x, & \cos z\end{array}\right], \quad S$$

the surface of $x^2+y^2 \leqq 4, \quad|z| \leqq 2$