/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Find the divergence of the follo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 14: Problem 14

Find the divergence of the following vector fields. $$\mathbf{F}=\left\langle e^{-x+y}, e^{-y+z}, e^{-z+x}\right\rangle$$

Short Answer

Expert verified
Question: Find the divergence of the vector field \(\mathbf{F}=\left\langle e^{-x+y}, e^{-y+z}, e^{-z+x}\right\rangle\). Answer: The divergence of the given vector field is \(\nabla \cdot \mathbf{F} = -e^{-x+y} - e^{-y+z} - e^{-z+x}\).

Step by step solution

01

Write down the divergence formula

The formula for the divergence of a vector field \(\mathbf{F}=\left\langle F_1, F_2, F_3 \right\rangle\) in Cartesian coordinates is given by: $$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}.$$ In this problem, we have the vector field: $$\mathbf{F}=\left\langle e^{-x+y}, e^{-y+z}, e^{-z+x}\right\rangle.$$
02

Calculate the partial derivatives

We need to find the partial derivatives of each component with respect to the corresponding variable: For \(F_1 = e^{-x+y}\): $$\frac{\partial F_1}{\partial x} = \frac{\partial}{\partial x} \left( e^{-x+y}\right) = -e^{-x+y}.$$ For \(F_2 = e^{-y+z}\): $$\frac{\partial F_2}{\partial y} = \frac{\partial}{\partial y} \left( e^{-y+z}\right) = -e^{-y+z}.$$ For \(F_3 = e^{-z+x}\): $$\frac{\partial F_3}{\partial z} = \frac{\partial}{\partial z} \left( e^{-z+x}\right) = -e^{-z+x}.$$
03

Calculate the divergence

Now, we'll plug these partial derivatives into the divergence formula: $$\nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z} = (-e^{-x+y}) + (-e^{-y+z}) + (-e^{-z+x}).$$ So, the divergence of the vector field is: $$\nabla \cdot \mathbf{F} = -e^{-x+y} - e^{-y+z} - e^{-z+x}.$$

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

Suppose an object with mass \(m\) moves in a region \(R\) in a conservative force field given by \(\mathbf{F}=-\nabla \varphi\) where \(\varphi\) is a potential function in a region \(R .\) The motion of the object is governed by Newton's Second Law of Motion, \(\mathbf{F}=m \mathbf{a}\) where a is the acceleration. Suppose the object moves from point \(A\) to point \(B\) in \(R\). a. Show that the equation of motion is \(m \frac{d \mathbf{v}}{d t}=-\nabla \varphi\) b. Show that \(\frac{d \mathbf{v}}{d t} \cdot \mathbf{v}=\frac{1}{2} \frac{d}{d t}(\mathbf{v} \cdot \mathbf{v})\) c. Take the dot product of both sides of the equation in part (a) with \(\mathbf{v}(t)=\mathbf{r}^{\prime}(t)\) and integrate along a curve between \(A\) and \(B\). Use part (b) and the fact that \(\mathbf{F}\) is conservative to show that the total energy (kinetic plus potential) \(\frac{1}{2} m|\mathbf{v}|^{2}+\varphi\) is the same at \(A\) and \(B\). Conclude that because \(A\) and \(B\) are arbitrary, energy is conserved in \(R\)

Prove the following identities. a. \(\iiint_{D} \nabla \times \mathbf{F} d V=\iint_{S}(\mathbf{n} \times \mathbf{F}) d S\) (Hint: Apply the Divergence Theorem to each component of the identity.) b. \(\iint_{S}(\mathbf{n} \times \nabla \varphi) d S=\oint_{C} \varphi d \mathbf{r}\) (Hint: Apply Stokes' Theorem to each component of the identity.)

Prove the following identities. Assume that \(\varphi\) is \(a\) differentiable scalar-valued function and \(\mathbf{F}\) and \(\mathbf{G}\) are differentiable vector fields, all defined on a region of \(\mathbb{R}^{3}\). $$\nabla \times(\nabla \times \mathbf{F})=\nabla(\nabla \cdot \mathbf{F})-(\nabla \cdot \nabla) \mathbf{F}$$

The potential function for the force field due to a charge \(q\) at the origin is \(\varphi=\frac{1}{4 \pi \varepsilon_{0}} \frac{q}{|\mathbf{r}|},\) where \(\mathbf{r}=\langle x, y, z\rangle\) is the position vector of a point in the field and \(\varepsilon_{0}\) is the permittivity of free space. a. Compute the force field \(\mathbf{F}=-\nabla \varphi\). b. Show that the field is irrotational; that is \(\nabla \times \mathbf{F}=\mathbf{0}\).

Consider the radial fields \(\mathbf{F}=\frac{\langle x, y, z\rangle}{\left(x^{2}+y^{2}+z^{2}\right)^{p / 2}}=\frac{\mathbf{r}}{|\mathbf{r}|^{p}},\) where \(p\) is a real number. Let \(S\) consist of the spheres \(A\) and \(B\) centered at the origin with radii \(0
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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.