Math Problems Vault

Problem 2.3.1

What is the volume enclosed by the ellipsoid

\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \, ?

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Problem 2.3.2

Evaluate

\iint _{\mathcal{A}} e^{-x^2-y^2} \, dx \, dy,

where $\mathcal{A} = \left\{ (x,y) \in \mathbb {R}^{2} \mid x^2+y^2 \leq 1\right\}$ .

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Problem 2.3.3

Given the fact that $\displaystyle \int_{-\infty }^{\infty } e^{-x^2} \, dx = \sqrt{\pi }$ , evaluate the integral

I = \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } e^{-(x^2+(y-x)^2+y^2)} \, dx \, dy.

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Problem 2.3.4

Evaluate

\iint _{\mathcal{R}} (x^3 - 3xy^2) \, dx \, dy,

where

\mathcal{R} = \left\{ (x,y) \in \mathbb {R}^{2} \mid (x+1)^2+y^2 \leq 9, \ (x-1)^2+y^2 \geq 1\right\} .

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Problem 2.3.5

(1)(2)

Let $\varphi (x,y)$ be a function with continuous second order partial derivatives such that

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(1)

$\varphi_{xx} + \varphi_{yy} + \varphi_x = 0$ in the punctured plane $\mathbb {R}^{2} \setminus \left\{ 0\right\}$ ,

(2)

$r\varphi_x \to \dfrac {x}{2\pi r}$ and $r\varphi_y \to \dfrac {y}{2\pi r}$ as $r = \sqrt{x^2+y^2} \to 0$ .

Let $C_R$ be the circle $x^2+y^2=R^2$ . Show that the line integral

\int _{C_R} e^x(-\varphi _y \, dx + \varphi _x \, dy)

is independent of $R$ , and evaluate it.

Problem 2.3.6

Let $S = \left\{ (x,y,z) \in \mathbb {R}^{3} \mid x^2+y^2+z^2=1\right\}$ denote the unit sphere in $\mathbb {R}^{3}$ . Evaluate the surface integral over $S$ :

\iint _S (x^2+y+z) \, dA.

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Problem 2.3.7

(1)(2)

Let $\overrightarrow {i}$ , $\overrightarrow {j}$ , and $\overrightarrow {k}$ be the usual unit vectors in $\mathbb {R}^{3}$ . Let $\overrightarrow {F}$ denote the vector field

(x^2+y-4)\overrightarrow {i} + 3xy\overrightarrow {j} + (2xz+z^2)\overrightarrow {k}.

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(1)

Compute $\nabla \times \overrightarrow {F}$ (the curl of $\overrightarrow {F}$ ).

(2)

Compute the integral of $\nabla \times \overrightarrow {F}$ over the surface $x^2+y^2+z^2=16$ , $z \geq 0$ .

Problem 2.3.8

Let the vector field $F$ in $\mathbb {R}^{3}$ have the form

F(r) = g(\left\| r\right\| ) r \qquad (r \neq (0,0,0)),

where $g$ is a real valued smooth function on $(0,\infty )$ and $\left\| \cdot \right\|$ denotes the Euclidean norm. ( $F$ is undefined at $(0,0,0)$ .) Prove that

\int _C F \cdot ds = 0

for any smooth closed path $C$ in $\mathbb {R}^{3}$ that does not pass through the origin.

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Problem 2.3.9

(1)(2)

Let $\mathcal{B}$ denote the unit ball of $\mathbb {R}^{3}$ , $\mathcal{B} = \left\{ r \in \mathbb {R}^{3} \mid \left\| r\right\| \leq 1\right\}$ . Let $\overrightarrow {J} = (J_1, J_2, J_3)$ be a smooth vector field on $\mathbb {R}^{3}$ that vanishes outside of $\mathcal{B}$ and satisfies $\nabla \cdot \overrightarrow {J} = 0$ .

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(1)

For $f$ a smooth, scalar-valued function defined on a neighborhood of $\mathcal{B}$ , prove that

\int _{\mathcal{B}} (\nabla f) \cdot \overrightarrow {J} \, dx \, dy \, dz = 0.

(2)

Prove that

\int _{\mathcal{B}} J_1 \, dx \, dy \, dz = 0.

Problem 2.3.10

Let $\mathcal{D}$ denote the open unit disc in $\mathbb {R}^{2}$ . Let $u$ be an eigenfunction for the Laplacian in $\mathcal{D}$ ; that is, a real valued function of class $C^2$ defined in $\overline{\mathcal{D}}$ , zero on the boundary of $\mathcal{D}$ but not identically zero, and satisfying the differential equation

\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = \lambda u,

where $\lambda$ is a constant. Prove that

\iint _{\mathcal{D}} \left|\operatorname {grad} u\right|^2 \, dx \, dy + \lambda \iint _{\mathcal{D}} u^2 \, dx \, dy = 0, \tag {$*$}

and hence that $\lambda < 0$ .

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Problem 2.3.11

Let $\lambda , a \in \mathbb {R}$ , with $a < 0$ . Let $u(x,y)$ be an infinitely differentiable function defined on an open neighborhood of closed unit disc $\mathcal{D}$ such that

\frac{\partial ^2 u}{\partial x^2} + \frac{\partial ^2 u}{\partial y^2} = \lambda u \qquad \text{in } \operatorname {int}(\mathcal{D})

D_n u = au \qquad \text{in } \partial \mathcal{D}.

Here $D_n u$ denotes the directional derivative of $u$ in the direction of the outward unit normal. Prove that if $u$ is not identically zero in the interior of $\mathcal{D}$ then $\lambda < 0$ .

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Problem 2.3.12

Let $f$ be a one-to-one $C^1$ map of $\mathbb {R}^{3}$ into $\mathbb {R}^{3}$ , and let $J$ denote its Jacobian determinant. Prove that if $x_0$ is any point of $\mathbb {R}^{3}$ and $Q_r(x_0)$ denotes the cube with center $x_0$ , side length $r$ , and edges parallel to the coordinate axes, then

\left|J(x_0)\right| = \lim _{r \to 0} r^{-3} \operatorname {vol}(f(Q_r(x_0))) \leq \limsup _{x \to x_0} \frac{\left\| f(x)-f(x_0)\right\| ^3}{\left\| x-x_0\right\| ^3}.

Here, $\left\| \cdot \right\|$ is the Euclidean norm in $\mathbb {R}^{3}$ .

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