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.) $w=x^5 y+x y^5, \quad R: x^2+y^2 \leqq 1, x \geqq 0, y \geqq 0$.

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.)

$$G=z, \quad S: x^2+y^2+z^2=9, \quad z \geqq 0$$

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 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.)

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

the straight-line segment from $(\pi, 1,0)$ to $\left(\frac{1}{2}, \pi, 1\right)$

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

$$\cos (x+y z)(d x+z d y+y d z)$$

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.) Hyperbolic cylinder $x^2-y^2=1$

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.)

$$\mathbf{F}=\left[\begin{array}{lll}4 x & x^2 y, & -x^2 z\end{array}\right], \quad S$$

the surface of the tetrahedron with vertices $(0,0,0),(1,0,0),(0,1,0),(0,0,1)$

Evaluate $\int_C f(\mathbf{r}) d s$ with $f$ and $C$ as follows. (Show the details.) $f=1+y^2+z^2, \quad$ C: $\mathbf{r}=[t, \cos t, \sin t], 0 \leqq t \leqq \pi$.

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.

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.)

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

, $\quad S$ the surface of $x^2+y^2 \leqq 9, \quad 0 \leqq z \leqq 2$

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.) Sphere $x^2+(y-1)^2+(z+2)^2=4$

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.)

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

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 $(-3,5)$

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