91Ó°ÊÓ

Describe the surface with the given parametric representation. $$\mathbf{r}(u, v)=\langle v \cos u, v \sin u, 4 v\rangle, \text { for } 0 \leq u \leq \pi, 0 \leq v \leq 3$$

Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces \(S\). \(\mathbf{F}=\langle x, 2 y, z\rangle ; S\) is the boundary of the tetrahedron in the first octant formed by the plane \(x+y+z=1\)

Calculate the divergence of the following radial fields. Express the result in terms of the position vector \(\mathbf{r}\) and its length \(|\mathbf{r}| .\) Check for agreement with Theorem 14.8. $$\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{4}}$$

Normal and tangential components Determine the points (if any) on the curve C at which the vector field \(\mathbf{F}\) is tangent to C and normal to C. Sketch C and a few representative vectors of \(\mathbf{F}\). $$\mathbf{F}=\langle y, x\rangle, \text { where } C=\left\\{(x, y): x^{2}+y^{2}=1\right\\}$$

Calculate the divergence of the following radial fields. Express the result in terms of the position vector \(\mathbf{r}\) and its length \(|\mathbf{r}| .\) Check for agreement with Theorem 14.8. $$\mathbf{F}=\langle x, y, z\rangle\left(x^{2}+y^{2}+z^{2}\right)=\mathbf{r}|\mathbf{r}|^{2}$$

Determine whether the following vector fields are conservative on the specified region. If so, determine a potential function. Let \(R^{*}\) and \(D^{*}\) be open regions of \(\mathbb{R}^{2}\) and \(\mathbb{R}^{3}\), respectively, that do not include the origin. $$\mathbf{F}=\langle y, x, 1\rangle \text { on } \mathbb{R}^{3}$$

Use the Divergence Theorem to compute the net outward flux of the following fields across the given surfaces \(S\). $$\mathbf{F}=\left\langle x^{2}, y^{2}, z^{2}\right\rangle ; S \text { is the sphere }\left\\{(x, y, z): x^{2}+y^{2}+z^{2}=25\right\\}$$

For the following velocity fields, compute the curl, make a sketch of the curl, and interpret the curl. $$v=\langle 0,0, y\rangle$$

Use a line integral on the boundary to find the area of the following regions. The region bounded by the parabolas \(\mathbf{r}(t)=\left\langle t, 2 t^{2}\right\rangle\) and \(\mathbf{r}(t)=\left\langle t, 12-t^{2}\right\rangle,\) for \(-2 \leq t \leq 2\)

Find the average value of the following functions on the given curves. \(f(x, y)=x+2 y\) on the line segment from \((1,1)\) to \((2,5).\)