# Assignment 11

Problem 1.  (a) Set up a triple integral (using $dV = dz\,dy\,dx$) that represents the volume of the solid region $S \subset \mathbb{R}^3$ bounded by the sphere $x^2+y^2+z^2 = 4$ and the cone $z = \sqrt{x^2+y^2}$.

(b) Change variables for your integral in part (a) so that you can compute the volume of $S$.

Problem 2.  For this problem we will use a function $F:\mathbb{R}^3\to\mathbb{R}^3$ that depends on three positive, constant numbers $a, b, c \in \mathbb{R}$:

$\displaystyle F(u,v,w) = (au, bv, cw)$.

(a) What is $F(1,0,0)$?  What is $F(0, 1, 0)$ and what is $F(0,0,1)$?  Describe in words and/or pictures what $F$ does to the unit sphere (centered at the origin).

(b) Compute the derivative matrix of $F$ and compute the determinant of this derivative matrix.

(c)  Compute the volume of the region bounded by the ellipsoid $E \subset \mathbb{R}^3$ given by the equation

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1.$

Problem 3.  Evaluate the (improper, one-dimensional) integral

$\displaystyle \int_{-\infty}^{\infty} \, e^{-x^2}\, dx.$

Problem 4.  Consider the helix curve $\gamma$ that is parameterized by the function $\vec{r}(t) = (\cos t, \sin t, t)$ where $t \in [0, 2\pi]$.

(a) Compute the length of the curve $\gamma$.

(b) If the function $f(x,y,z) = x^2-y^2+z^2$ measures the density at every point $(x,y,z)$ on $\gamma$, compute the total mass of $\gamma$.

(c) Compute the amount of work done by the vector field $F(x,y,z) = (-y, x, z/(x^2+y^2)\,)$ in moving a particle along $\gamma$.

Problem 5.  Let $f:\mathbb{R}^3\to\mathbb{R}$ be any differentiable function.  Prove / compute that

$\displaystyle \nabla \times \nabla f = \text{curl}\,\left(\nabla f\right) = \vec{0}.$

Problem 6.  (a) Compute the line integral

$\displaystyle \int_{\gamma} \, F\cdot d\vec{s}$

where $F(x,y) = (ye^{xy}, xe^{xy})$ and $\gamma$ is the ellipse (traversed in a counter-clockwise direction) given by

$\displaystyle \frac{x^2}{16} + \frac{y^2}{9} = 1$.

(b) Compute the line integral

$\displaystyle \int_{\alpha} \, F\cdot d\vec{s}$

where $F$ is the same vector field from part (a), but $\alpha$ is only the top half of the ellipse from part (a) (also given a counter-clockwise orientation).

Problem 7.  State the following integral theorems:

The Gradient Theorem, Greens’ Theorem, Stokes’ Theorem, the Divergence Theorem.

Problem 8.  Use Greens Theorem to compute the area of the region $D \subset \mathbb{R}^2$ given by

$\displaystyle D = \left\{ (x,y) : \frac{x^2}{4} + \frac{y^2}{25} \leq 1 \right\}.$

Problem 9.  Set up a double integral that represents the surface area of the graph for $f(x,y) = x^2+y^2$ over the domain $\left\{(x,y) : x^2+y^2 \leq 4\right\}$.

Problem 10.  (a) Write down the meaning of the following symbols:

$ds =$

$d\vec{s} =$

$dS =$

$d\vec{S} =$

(b) Which of the above symbols are used for line integrals?  Which are used for surface integrals?

(c) Given a surface $S \subset \mathbb{R}^3$, write down an integral expression that represents its surface area (assuming the surface has been parameterized by a function $\vec{r}(s, t) : D \to \mathbb{R}^3$).  In addition, write down an expression for a unit normal to $S$.

Problem 11.  Compute the flux of the vector field $F(x,y,z) = (x, y, z)$ out of the unit sphere $S = \left\{(x,y,z) : x^2+y^2+z^2 = 1\right\}$.

Problem 12.  (a) Compute the flux of the vector field $F(x,y,z) = (-y,x,z)$ out of the cylindrical surface

$\displaystyle S = \left\{(x,y,z) : x^2+y^2 = 1 \text{ and } -1 \leq z \leq 1\right\}$.

(b) Draw a picture of the surface $S$.  Is this surface the boundary of a solid region?  Why or why not?

(c) Describe / draw a picture of the boundary of $S$ or explain why it has no boundary.  If $S$ does have a boundary, how should it be oriented so as to be consistent with the outward normal $n$?

(d) Compute the flux

$\displaystyle \iint_S \, \left(\nabla \times F\right)\,\cdot\,d\vec{S}$.

Problem 13.  Suppose $W \subset \mathbb{R}^3$ is a solid region that has volume $\text{Vol}(W) = 10$.  If $S = \partial W$ is the boundary surface of $W$, compute the flux of $F$ out of the surface $S$ where

$F(x,y,z) = (x+y^2+z^5, \cos(xz)e^z + y, xy + \arctan(xy) + z)$