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)

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