Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 9: Problem 1

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. \(\mathbf{u}=5 \mathbf{i}+2 \mathbf{j}-3 \mathbf{k}, \mathbf{v}=\mathbf{i}-4 \mathbf{j}+2 \mathbf{k} .\)

Short Answer

Expert verified
The cross product for each pair of vectors using both methods should match, providing practice in both calculation methods and verification of the results.

Step by step solution

01

Compute Cross Product Using Permutation Symbol

The permutation symbol \(ε_{ijk}\) is a mathematical function used in the computation of cross products. It's equal to +1 if \(ijk\) is an even permutation of 123, -1 if it's an odd permutation, and 0 if any two indexes are the same. Use this formula: (\(\mathbf{u} \times \mathbf{v}\))_i = Σ_{j,k} ε_{ijk} u_j v_k . Do this for each part (a, b, c) in turn.
02

Verify Answer Using Traditional Methods

The traditional method for the cross product involves finding the determinant of a 3x3 matrix. Use this formula: \(\mathbf{u} \times \mathbf{v} = \mathbf{i}(u_2v_3 - u_3v_2) - \mathbf{j}(u_1v_3 - u_3v_1) + \mathbf{k}(u_1v_2 - u_2v_1)\). Insert the corresponding components of vectors \(\mathbf{u}\) and \(\mathbf{v}\) from each part (a, b, c) in the formula and calculate the determinant.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

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 \(\mathbf{r}\).

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 \(x+y+z=1\)

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: $$ \frac{1}{2} v^{2}+\phi+\frac{p}{\rho}=\text { const. } $$

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: $$ \begin{gathered} \nabla f=\frac{\partial f}{\partial r} \hat{\mathbf{e}}_{r}+\frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\mathbf{e}}_{\theta}+\frac{\partial f}{\partial z} \hat{\mathbf{e}}_{z} \\ \nabla \cdot \mathbf{F}=\frac{1}{r} \frac{\partial\left(r F_{r}\right)}{\partial r}+\frac{1}{r} \frac{\partial F_{\theta}}{\partial \theta}+\frac{\partial F_{z}}{\partial z} \\ \nabla \times \mathbf{F}=\left(\frac{1}{r} \frac{\partial F_{z}}{\partial \theta}-\frac{\partial F_{\theta}}{\partial z}\right) \hat{\mathbf{e}}_{r}+\left(\frac{\partial F_{r}}{\partial z}-\frac{\partial F_{z}}{\partial r}\right) \hat{\mathbf{e}}_{\theta}+\frac{1}{r}\left(\frac{\partial\left(r F_{\theta}\right)}{\partial r}-\frac{\partial F_{r}}{\partial \theta}\right) \\\ \nabla^{2} f=\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial f}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} f}{\partial \theta^{2}}+\frac{\partial^{2} f}{\partial z^{2}}. \end{gathered} $$

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. \(\left|\begin{array}{ccc}1 & 2 & 2 \\ 4 & -6 & 3 \\ 2 & 3 & 1\end{array}\right|\)
See all solutions

Recommended explanations on Combined Science Textbooks

Synergy

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free