Chapter 9

Compute \(\mathbf{u} \times \mathbf{v}\) using the permutation symbol. Verify your answer by computing these products using traditional methods. a. \(\mathbf{u}=2 \mathbf{i}-3 \mathbf{k}, \mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\). b. \(\mathbf{u}=\mathbf{i}+\mathbf{j}+\mathbf{k}, \mathbf{v}=\mathbf{i}-\mathbf{k}\). c. \(\mathbf{u}=5 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \mathbf{v}=\mathbf{i}-4 \mathbf{j}+2 \mathbf{k} .\)

Compute the following determinants using the permutation symbol. Verify your answer. a. \(\left|\begin{array}{ccc}3 & 2 & 0 \\ 1 & 4 & -2 \\ -1 & 4 & 3\end{array}\right|\) b. \(\left|\begin{array}{ccc}1 & 2 & 2 \\ 4 & -6 & 3 \\ 2 & 3 & 1\end{array}\right|\)

Prove the following vector identities: a. \((\mathbf{a} \times \mathbf{b}) \cdot(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{c})(\mathbf{b} \cdot \mathbf{d})-(\mathbf{a} \cdot \mathbf{d})(\mathbf{b} \cdot \mathbf{c})\) b. \((\mathbf{a} \times \mathbf{b}) \times(\mathbf{c} \times \mathbf{d})=(\mathbf{a} \cdot \mathbf{b} \times \mathbf{d}) \mathbf{c}-(\mathbf{a} \cdot \mathbf{b} \times \mathbf{c}) \mathbf{d}\).

A particle moves on a straight line, \(\mathbf{r}=t \mathbf{u}\), from the center of a disk. If the disk is rotating with angular velocity \(\omega\), then \(\mathbf{u}\) rotates. Let \(\mathbf{u}=\) \((\cos \omega t) \mathbf{i}+(\sin \omega t) \mathbf{j}\) a. Determine the velocity, \(\mathbf{v}\). b. Determine the acceleration, a. c. Describe the resulting acceleration terms identifying the centripetal acceleration and Coriolis acceleration.

Compute the gradient of the following: a. \(f(x, y)=x^{2}-y^{2}\) b. \(f(x, y, z)=y z+x y+x z\). c. \(f(x, y)=\tan ^{-1}\left(\frac{y}{x}\right)\). d. \(f(x, y, z)=2 y^{x} \cos z-5 \sin z \cos y\).

Find the directional derivative of the given function at the indicated point in the given direction. a. \(f(x, y)=x^{2}-y^{2},(3,2), \mathbf{u}=\mathbf{i}+\mathbf{j}\). b. \(f(x, y)=\frac{y}{x},(2,1), \mathbf{u}=3 \mathbf{i}+4 \mathbf{j}\) c. \(f(x, y, z)=x^{2}+y^{2}+z^{2},(1,0,2), \mathbf{u}=2 \mathbf{i}-\mathbf{j}\).

Zaphod Beeblebrox was in trouble after the infinite improbability drive caused the Heart of Gold, the spaceship Zaphod had stolen when he was President of the Galaxy, to appear between a small insignificant planet and its hot sun. The temperature of the ship's hull is given by \(T(x, y, z)=\) \(e^{-k\left(x^{2}+y^{2}+z^{2}\right)}\) Nivleks. He is currently at \((1,1,1)\), in units of globs, and \(k=2\) globs \(^{-2}\). (Check the Hitchhikers Guide for the current conversion of globs to kilometers and Nivleks to Kelvin.) a. In what direction should he proceed so as to decrease the temperature the quickest? b. If the Heart of Gold travels at \(e^{6}\) globs per second, then how fast will the temperature decrease in the direction of fastest decline?

A particle moves under the force field \(\mathbf{F}=-\nabla V\), where the potential function is given by \(V(x, y, z)=x^{3}+y^{3}-3 x y+5\). Find the equilibrium points of \(\mathbf{F}\) and determine if the equilibria are stable or unstable.

For the given vector field, find the divergence and curl of the field. a. \(\mathbf{F}=x \mathbf{i}+y \mathbf{j}\) b. \(\mathbf{F}=\frac{y}{r} \mathbf{i}-\frac{x}{r} \mathbf{j}\), for \(r=\sqrt{x^{2}+y^{2}}\). c. \(\mathbf{F}=x^{2} y \mathbf{i}+z \mathbf{j}+x y z \mathbf{k} .\)

Write the following using \(\epsilon_{i j k}\) notation and simplify if possible. a. \(\mathbf{C} \times(\mathbf{A} \times(\mathbf{A} \times \mathbf{C}))\) b. \(\nabla \times(\nabla \times \mathbf{A})\) c. \(\nabla \times \nabla \phi\).

Prove the identities: a. \(\nabla \cdot(\nabla \times \mathbf{A})=0\). b. \(\nabla \cdot(f \nabla g-g \nabla f)=f \nabla^{2} g-g \nabla^{2} f\). c. \(\nabla r^{n}=n r^{n-2} \mathbf{r}, \quad n \geq 2\)

For \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\) and \(r=|\mathbf{r}|\), simplify the following. a. \(\nabla \times(\mathbf{k} \times \mathbf{r})\). b. \(\nabla \cdot\left(\frac{\mathrm{r}}{r}\right)\). c. \(\nabla \times\left(\frac{\mathbf{r}}{r}\right)\). d. \(\nabla \cdot\left(\frac{\mathrm{r}}{r^{3}}\right)\). e. \(\nabla \times\left(\frac{\mathrm{r}}{r^{3}}\right)\).

Newton's Law of Gravitation gives the gravitational force between two masses as $$ \mathbf{F}=-\frac{G m M}{r^{3}} \mathbf{r} $$ a. Prove that \(\mathbf{F}\) is irrotational. b. Find a scalar potential for \(\mathbf{F}\).

Consider a constant electric dipole moment \(\mathrm{p}\) at the origin. It produces an electric potential of \(\phi=\frac{\mathrm{p} \cdot \mathrm{r}}{4 \pi \epsilon_{0} r^{3}}\) outside the dipole. Noting that \(\mathbf{E}=-\nabla \phi\), find the electric field at \(\mathbf{r}\).

In fluid dynamics, the Euler equations govern inviscid fluid flow and provide quantitative statements on the conservation of mass, momentum, and energy. The continuity equation is given by $$ \frac{\partial \rho}{\partial t}+\nabla \cdot(\rho \mathbf{v})=0 $$ where \(\rho(x, y, z, t)\) is the mass density and \(\mathbf{v}(x, y, z, t)\) is the fluid velocity. The momentum equations are given by $$ \frac{\partial \rho \mathbf{v}}{\partial t}+\mathbf{v} \cdot \nabla(\rho \mathbf{v})=\mathbf{f}-\nabla p $$ Here, \(p(x, y, z, t)\) is the pressure and \(\mathbf{f}\) is the external force per volume. a. Show that the continuity equation can be rewritten as $$ \frac{\partial \rho}{\partial t}+\rho \nabla \cdot(\mathbf{v})+\mathbf{v} \cdot \nabla \rho=0 $$ b. Prove the identity \(\frac{1}{2} \nabla v^{2}=\mathbf{v} \cdot \nabla \mathbf{v}\) for irrotational \(\mathbf{v}\). c. Assume that \- the external forces are conservative \((\mathbf{f}=-\rho \nabla \phi)\), \- the velocity field is irrotational \((\nabla \times \mathbf{v}=\mathbf{0})\), \- the fluid is incompressible \((\rho=\) const \()\), and \- the flow is steady, \(\frac{\partial \mathrm{v}}{\partial t}=0\). Under these assumptions, prove Bernoulli's Principle: $$ \frac{1}{2} v^{2}+\phi+\frac{p}{\rho}=\text { const. } $$

Find the lengths of the following curves: a. \(y(x)=x\) for \(x \in[0,2]\). b. \((x, y, z)=(t, \ln t, 2 \sqrt{2} t)\) for \(1 \leq t \leq 2\). c. \(y(x)=\cosh x, x \in[-2,2]\). (Recall the hanging chain example from classical dynamics.)

Consider the integral \(\int_{C} y^{2} d x-2 x^{2} d y\). Evaluate this integral for the following curves: a. \(C\) is a straight line from \((0,2)\) to \((1,1)\). b. \(C\) is the parabolic curve \(y=x^{2}\) from \((0,0)\) to \((2,4)\). c. \(C\) is the circular path from \((1,0)\) to \((0,1)\) in a clockwise direction.

Evaluate \(\int_{C}\left(x^{2}-2 x y+y^{2}\right) d s\) for the curve \(x(t)=2 \cos t, y(t)=2 \sin t\), \(0 \leq t \leq \pi\)

Prove that the magnetic flux density, B, satisfies the wave equation.

Let \(C\) be a closed curve and \(D\) the enclosed region. Prove the identity $$ \int_{C} \phi \nabla \phi \cdot \mathbf{n} d s=\int_{D}\left(\phi \nabla^{2} \phi+\nabla \phi \cdot \nabla \phi\right) d A. $$

Let \(S\) be a closed surface and \(V\) the enclosed volume. Prove Green's first and second identities, respectively. a. \(\int_{S} \phi \nabla \psi \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi+\nabla \phi \cdot \nabla \psi\right) d V\). b. \(\int_{S}[\phi \nabla \psi-\psi \nabla \phi] \cdot \mathbf{n} d S=\int_{V}\left(\phi \nabla^{2} \psi-\psi \nabla^{2} \phi\right) d V\).

Let \(C\) be a closed curve and \(D\) the enclosed region. Prove Green's identities in two dimensions. a. First prove $$ \int_{D}(v \nabla \cdot \mathbf{F}+\mathbf{F} \cdot \nabla v) d A=\int_{C}(v \mathbf{F}) \cdot d \mathbf{s} $$ b. Let \(\mathbf{F}=\nabla u\) and obtain Green's first identity, $$ \int_{D}\left(v \nabla^{2} u+\nabla u \cdot \nabla v\right) d A=\int_{C}(v \nabla u) \cdot d \mathbf{s} $$ c. Use Green's first identity to prove Green's second identity, $$ \int_{D}\left(u \nabla^{2} v-v \nabla^{2} u\right) d A=\int_{C}(u \nabla v-v \nabla u) \cdot d \mathbf{s} $$

Compute the following integrals: a. \(\int_{C}\left(x^{2}+y\right) d x+\left(3 x+y^{3}\right) d y\) for \(C\) the ellipse \(x^{2}+4 y^{2}=4\). b. \(\int_{S}(x-y) d y d z+\left(y^{2}+z^{2}\right) d z d x+\left(y-x^{2}\right) d x d y\) for \(S\) the positively oriented unit sphere. c. \(\int_{C}(y-z) d x+(3 x+z) d y+(x+2 y) d z\), where \(C\) is the curve of intersection between \(z=4-x^{2}-y^{2}\) and the plane \(x+y+z=0\). d. \(\int_{C} x^{2} y d x-x y^{2} d y\) for \(C\) a circle of radius 2 centered about the origin. e. \(\int_{S} x^{2} y d y d z+3 y^{2} d z d x-2 x z^{2} d x d y\), where \(S\) is the surface of the cube \([-1,1] \times[-1,1] \times[-1,1]\).

Use Stokes' Theorem to evaluate the integral $$ \int_{C}-y^{3} d x+x^{3} d y-z^{3} d z $$ for \(C\) the (positively oriented) curve of intersection between the cylinder \(x^{2}+y^{2}=1\) and the plane \(x+y+z=1\)

Use Stokes' Theorem to derive the integral form of Faraday's law, $$ \int_{C} \mathbf{E} \cdot d \mathbf{s}=-\frac{\partial}{\partial t} \iint_{D} \mathbf{B} \cdot d \mathbf{S} $$ from the differential form of Maxwell's equations.

For cylindrical coordinates, $$ \begin{aligned} x &=r \cos \theta \\ y &=r \sin \theta \\ z &=z \end{aligned} $$ find the scale factors and derive the following expressions: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial r} \hat{\mathbf{e}}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{\partial f}{\partial z} \hat{\mathbf{e}}_{z} \\ \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial\left(r F_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z} \\ \nabla \times \mathbf{F}=\left(\frac{1}{r} \frac{\partial F_{z}}{\partial \theta}-\frac{\partial F_{\theta}}{\partial z}\right) \hat{\mathbf{e}}_{r}+\left(\frac{\partial F_{r}}{\partial z}-\frac{\partial F_{z}}{\partial r}\right) \hat{\mathbf{e}}_{\theta}+\frac{1}{r}\left(\frac{\partial\left(r F_{\theta}\right)}{\partial r}-\frac{\partial F_{r}}{\partial \theta}\right) \\\ \nabla^{2} f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}. \end{gathered} $$

For spherical coordinates, $$ \begin{aligned} x &=\rho \sin \theta \cos \phi \\ y &=\rho \sin \theta \sin \phi \\ z &=\rho \cos \theta \end{aligned} $$ find the scale factors and derive the following expressions: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial \rho} \hat{\mathbf{e}}_{\rho}+\frac{1}{\rho} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{1}{\rho \sin \theta} \frac{\partial f}{\partial \phi} \hat{\mathbf{e}}_{\phi} \\ \nabla \cdot \mathbf{F}=\frac{1}{\rho^{2}} \frac{\partial\left(\rho^{2} F_{\rho}\right)}{\partial \rho}+\frac{1}{\rho \sin \theta} \frac{\partial\left(\sin \theta F_{\theta}\right)}{\partial \theta}+\frac{1}{\rho \sin \theta} \frac{\partial F_{\phi}}{\partial \phi}. \end{gathered} $$ $$ \begin{aligned} \nabla \times \mathbf{F}=& \frac{1}{\rho \sin \theta}\left(\frac{\partial\left(\sin \theta F_{\phi}\right)}{\partial \theta}-\frac{\partial F_{\theta}}{\partial \phi}\right) \hat{\mathbf{e}}_{\rho}+\frac{1}{\rho}\left(\frac{1}{\sin \theta} \frac{\partial F_{\rho}}{\partial \phi}-\frac{\partial\left(\rho F_{\phi}\right)}{\partial \rho}\right) \\ &+\frac{1}{\rho}\left(\frac{\partial\left(\rho F_{\theta}\right)}{\partial \rho}-\frac{\partial F_{\rho}}{\partial \theta}\right) \hat{\mathbf{e}}_{\phi} \end{aligned} $$ $$ \nabla^{2} f=\frac{1}{\rho^{2}} \frac{\partial}{\partial \rho}\left(\rho^{2} \frac{\partial f}{\partial \rho}\right)+\frac{1}{\rho^{2} \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial f}{\partial \theta}\right)+\frac{1}{\rho^{2} \sin ^{2} \theta} \frac{\partial^{2} f}{\partial \phi^{2}}. $$

The moments of inertia for a system of point masses are given by sums instead of integrals. For example, \(I_{x x}=\sum_{i} m_{i}\left(y_{i}^{2}+z_{i}^{2}\right)\) and \(I_{x y}=\) \(-\sum_{i} m_{i} x_{i} y_{i}\). Find the inertia tensor about the origin for \(m_{1}=2.0 \mathrm{~kg}\) at \((1.0,0,1.0), m_{2}=5.0 \mathrm{~kg}\) at \((1.0,-1.0,0)\), and \(m_{3}=1.0 \mathrm{~kg}\) at \((1.0,1.0,1.0)\) where the coordinate units are in meters.

Consider the octant of a uniform sphere of density 5 grams per cubic centimeter and radius \(a\) lying in the first octant. a. Find the inertia tensor about the origin. b. What are the principal moments of inertia? c Find the principal axes of inertia, that is, the eigenvectors of the inertia tensor.

Let \(T^{\alpha}\) be a contravariant vector and \(S_{\alpha}\) be a covariant vector. a. Show that \(R_{\beta}=g_{\alpha \beta} T^{\alpha}\) is a covariant vector. b. Show that \(R^{\beta}=g^{\alpha \beta} S_{\alpha}\) is a contravariant vector.

Show that \(T^{\alpha \beta \gamma \delta \rho} S_{\beta \rho}\) is a tensor. What is its rank?

The line element in terms of the metric tensor, \(g_{\alpha \beta}\) is given by $$ d s^{2}=g_{\alpha \beta} d x^{\alpha} d x^{\beta}. $$ Show that the transformed metric for the transformation \(x^{\prime \alpha}=x^{\prime \alpha}\left(x^{\beta}\right)\) is given by $$ g_{\gamma \delta}^{\prime}=g_{\alpha \beta} \frac{\partial x^{\alpha}}{\partial x^{\prime} \gamma} \frac{\partial x^{\beta}}{\partial x^{\prime \delta}} $$