Homework 13

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\dydx}{\frac{dy}{dx}} \newcommand{\div}{\operatorname{div}} \newcommand{\curl}{\operatorname{curl}} \newcommand{\proj}{\textrm{proj}} % For boldface vectors: \renewcommand{\vec}[1]{\mathbf{#1}} $$

1. Integrate $g(x, y, z) = x\sqrt{y^2+4}$ over the surface $S$ that is the portion of the surface $y^2 + 4z = 16$ that lies between the planes $x = 0$, $x = 1$, and $z = 0$.

2. Find the outward flux of the field $\vec F = xz\, \vec i + yz \,\vec j + \vec k$ across the surface of the portion of the sphere $x^2 + y^2 + z^2 = 25$ above the plane $z = 3$.

3. Let $\vec n$ be the outer unit normal of the surface $S$ given by $4x^2 + 9y^2 + 36z^2 = 26, z \geq 0$, and let $\vec F(x, y, z) = y\vec i + x^2 \vec j + (x^2 + y^4)^{3/2} \sin(e^{\sqrt{xyz}})\vec k$.

Find the value of $$ \iint_S \curl \vec F \cdot d \vec S. $$

Hint: One parametrization of the ellipse at the base of the shell is of the form $x = a \cos(t), y = b\cos(t)$, for some constants $a, b$.

Further Hint: That last compnent of $\vec F$ is so complicated, it must not matter. Why not?

4. Use the Divergence Theorem to find the outward flux of $\vec F$ across the boundary of the region $D$, where $\vec F(x, y, z) = y\, \vec i + xy\, \vec j - z\, \vec k$ and $D$ is the region inside the solid cylinder $x^2 + y^2 \leq 4$, between the plane $z = 0$ and the paraboloid $z = x^2 + y^2$.

5. 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 $\vec F(x, y, z) = (-x^2 - 4xy) \,\vec i - 6 yz\, \vec j + 12z\, \vec k$ outward through the six sides is the greatest. What is the value of the greatest flux?

6. Consider the surface $S$ consisteng of the part of the cylinder $y^2+z^2=1$ where $z\geq 0$ and $1\leq x \leq 3$. Find the flux of the vector field $$\vec F(x,y,z) = \tan^{-1}\frac{y}{z+1}\,\vec i + \,x^2\vec j + z\,\vec k $$ upward through $S$ indirectly as follows*.

• Make $S$ into a closed surface by capping its ends with half-discs and its bottom with a rectangle.
• Find the flux through each of these "new" surfaces without actually computing a double integral (i.e., use $\iint_S \vec F\cdot\vec n\,dS$).
• Apply the divergence theorem and draw a conclusion about the flux through the original surface $S$.

*Following this procedure, one should not need to take a single antiderivative.