91Ó°ÊÓ

Use the surface integral in Stokes" Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\). $$\begin{aligned} &\mathbf{F}=(y-z) \mathbf{i}+(z-x) \mathbf{j}+(x+z) \mathbf{k}\\\ &S: \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+\left(9-r^{2}\right) \mathbf{k}\\\ &0 \leq r \leq 3, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Use the surface integral in Stokes" Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n}\). $$\begin{aligned} &\mathbf{F}=x^{2} y \mathbf{i}+2 y^{3} \mathbf{z} \mathbf{j}+3 z \mathbf{k}\\\ &\text { S: } \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+r \mathbf{k}\\\ &0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi \end{aligned}$$

Integrate \(G(x, y, z)=z-x\) over the portion of the graph of \(z=x+y^{2}\) above the triangle in the \(x y\) -plane having vertices \((0,0,0),(1,1,0),\) and \((0,1,0),\) (See accompanying figure.)

Find the work done by \(\mathbf{F}\) in moving a particle once counterclockwise around the given curve. \(\mathbf{F}=(4 x-2 y) \mathbf{i}+(2 x-4 y) \mathbf{j}\) C: The circle \((x-2)^{2}+(y-2)^{2}=4\)

Zero circulation Use Equation (8) and Stokes" Theorem to show that the circulations of the following fields around the boundary of any smooth orientable surface in space are zero. a. \(\mathbf{F}=2 x \mathbf{i}+2 y \mathbf{j}+2 z \mathbf{k}\) b. \(\mathbf{F}=\nabla\left(x y^{2} z^{3}\right)\) d. \(\mathbf{F}=\nabla f\) c. \(\mathbf{F}=\nabla \times(x \mathbf{i}+y \mathbf{j}+z \mathbf{k})\)

Use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the specified direction. Sphere \(\quad \mathbf{F}=z \mathbf{k}\) across the portion of the sphere \(x^{2}+y^{2}+\) \(z^{2}=a^{2}\) in the first octant in the direction away from the origin.

Maximum flux Among all rectangular solids defined by the inequalities \(0 \leq x \leq a, 0 \leq y \leq b, 0 \leq z \leq 1,\) find the one for which the total flux of \(\mathbf{F}=\left(-x^{2}-4 x y\right) \mathbf{i}-6 y z \mathbf{j}+12 z \mathbf{k}\) out- ward through the six sides is greatest. What is the greatest flux?

Let \(F\) be a differentiable vector field and let \(g(x, y, z)\) be a differentiable scalar function. Verify the following identities. a. \(\nabla \cdot(g \mathbf{F})=g \nabla \cdot \mathbf{F}+\nabla g \cdot \mathbf{F}\) b. \(\nabla \times(g \mathbf{F})=g \nabla \times \mathbf{F}+\nabla g \times \mathbf{F}\)

Show that the values of the integrals do not depend on the path taken from \(A\) to \(B\). $$\int_{A}^{B} \frac{x d x+y d y+z d z}{\sqrt{x^{2}+y^{2}+z^{2}}}$$

Zero curl, yet the field is not conservative Show that the curl of $$ \mathbf{F}=\frac{-y}{x^{2}+y^{2}} \mathbf{i}+\frac{x}{x^{2}+y^{2}} \mathbf{j}+z \mathbf{k} $$ is zero but that $$ \oint_{c} \mathbf{F} \cdot d \mathbf{r} $$ is not zero if \(C\) is the circle \(x^{2}+y^{2}=1\) in the \(x y\) -plane. (Theorem 7 does not apply here because the domain of \(\mathbf{F}\) is not simply connected. The field \(\mathbf{F}\) is not defined along the \(z\) -axis so there is no way to contract \(C\) to a point without leaving the domain of \(\mathbf{F}\).)