/*! 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 12 In Exercises \(7-12,\) find a po... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 16: Problem 12

In Exercises \(7-12,\) find a potential function \(f\) for the field \(\mathbf{F}\) $$ \begin{array}{l}{\mathbf{F}=\frac{y}{1+x^{2} y^{2}} \mathbf{i}+\left(\frac{x}{1+x^{2} y^{2}}+\frac{z}{\sqrt{1-y^{2} z^{2}}}\right) \mathbf{j}+} \\ {\quad\left(\frac{y}{\sqrt{1-y^{2} z^{2}}}+\frac{1}{z}\right) \mathbf{k}}\end{array} $$

Short Answer

Expert verified
The potential function is \( f(x, y, z) = \frac{1}{y} \tan^{-1}(xy) + \sin^{-1}(yz) + \log z + C \).

Step by step solution

01

Verify if the Field is Conservative

To determine if there's a potential function \( f \) for the field \( \mathbf{F} \), the field must be conservative, which means its curl must be zero. Calculate the curl of \( \mathbf{F} \) and confirm if it's zero: \( abla \times \mathbf{F} = \vec{0} \). If \( abla \times \mathbf{F} = \vec{0} \) everywhere, then \( \mathbf{F} \) is conservative.
02

Integrate the First Component Respect to x

Since \( \mathbf{F} \) is conservative, find a potential function \( f \) by first integrating the \( x \)-component of \( \mathbf{F} \), \( \frac{y}{1+x^2 y^2} \), with respect to \( x \), treating \( y \) and \( z \) as constants. This gives: \[ f(x, y, z) = \int \frac{y}{1+x^2 y^2} \, dx + g(y, z)\]Hence, \( f(x, y, z) = \frac{1}{y} \tan^{-1}(xy) + g(y, z) \).
03

Differentiate with Respect to y and Solve

To find \( g(y, z) \), differentiate \( f \) with respect to \( y \), and set this equal to the \( y \)-component of \( \mathbf{F} \):\[ \frac{\partial}{\partial y} \left( \frac{1}{y} \tan^{-1}(xy) + g(y, z) \right) = \frac{x}{1+x^2 y^2} + \frac{z}{\sqrt{1-y^2 z^2}}\]Solving gives:\[ -\frac{\tan^{-1}(xy)}{y^2} + \frac{x^2}{1+x^2 y^2} = \frac{x}{1+x^2 y^2} \, herefore \, g(y,z) = \int \frac{z}{\sqrt{1-y^2 z^2}} \, dy \]Integrating the term \( \frac{z}{\sqrt{1-y^2 z^2}} \) gives \( g(y,z) = \sin^{-1}(yz) + h(z) \).
04

Integrate Remaining Component with Respect to z

In the last step, check consistency with the \(z\) component by differentiating \( h(z) \):\[ \frac{\partial}{\partial z}(\sin^{-1}(yz) + h(z)) = \frac{y}{\sqrt{1-y^2 z^2}} + \frac{1}{z}\]Assure the equation is satisfied by defining \( h(z) = \log z + C \), where \( C \) is any constant.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conservative Fields
In vector calculus, a field is called conservative if it can be represented as the gradient of a scalar potential function. The significance of a conservative field lies in the fact that the line integral between two points is independent of the path taken. This is an essential property when working with potential energy and various physics applications.
To determine if a vector field is conservative, we typically check its curl. If the curl of the vector field is zero, the field is deemed conservative. Thus, we can say a vector field \( \mathbf{F} \) is conservative if:
  • \( abla \times \mathbf{F} = \vec{0} \).
When this condition is met, it guarantees the existence of a potential function \( f \) for the field.
Potential Functions
Potential functions are scalar functions whose gradients give us the original vector field. In other words, if \( \mathbf{F} = abla f \), then \( f \) is a potential function for \( \mathbf{F} \).
When a vector field is conservative, finding a potential function can simplify many calculations, especially when evaluating integrals. To construct a potential function, one typically:
  • Integrates each component of the vector field.
  • Combines results and determines arbitrary functions of the other variables.
These steps, as seen in the solution, begin with integrating the \( x \)-component of \( \mathbf{F} \) to form part of the potential function. Subsequent steps involve solving for additional terms derived from integrating with respect to other variables.
Curl of a Vector Field
The curl of a vector field measures the field's tendency to rotate around a point. It is a vector operation that results in a new vector. For a vector field \( \mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k} \), the curl is expressed as:
  • \( abla \times \mathbf{F} = \left( \frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z} \right)\mathbf{i} + \left( \frac{\partial P}{\partial z} - \frac{\partial R}{\partial x} \right)\mathbf{j} + \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\mathbf{k} \).
For a vector field to be conservative, the curl must be zero. Checking the curl helps confirm whether or not the potential function can exist. As shown in the step-by-step solution, verifying that the curl equals zero was the first and crucial step in establishing the conservativeness of \( \mathbf{F} \).
Integration Techniques
Integration is used extensively when constructing a potential function from a conservative vector field. When integrating, it's essential to treat other variables as constants. For example:
  • Integrating the \( x \)-component involves treating \( y \) and \( z \) as constants.
  • Finding functions of the remaining variables and incorporating them alongside constants is crucial when integrating further components.
Each step of integration might introduce additional functions of independent variables other than the one being integrated over.
In our example, after integrating with respect to \( x \), results often depend on \( y \) and \( z \). Thus, further integration or differentiation with respect to these variables is needed to refine the potential function, as shown with functions \( g(y,z) \) and \( h(z) \) in the solution.

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

Zero circulation Let \(f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{-1 / 2} .\) Show that the clockwise circulation of the field \(\mathbf{F}=\nabla f\) around the circle \(x^{2}+y^{2}=a^{2}\) in the \(x y\) -plane is zero a. by taking \(\mathbf{r}=(a \cos t) \mathbf{i}+(a \sin t) \mathbf{j}, 0 \leq t \leq 2 \pi,\) and integrating \(\mathbf{F} \cdot d \mathbf{r}\) over the circle. b. by applying Stokes' Theorem.

In Exercises \(13-18\) , use the surface integral in Stokes' Theorem to calculate the flux of the curl of the field \(\mathbf{F}\) across the surface \(S\) in the direction of the outward unit normal \(\mathbf{n} .\) $$ \begin{array}{l}{\mathbf{F}=x^{2} y \mathbf{i}+2 y^{3} z \mathbf{j}+3 z \mathbf{k}} \\ {S : \quad \mathbf{r}(r, \theta)=(r \cos \theta) \mathbf{i}+(r \sin \theta) \mathbf{j}+r \mathbf{k}} \\ {0 \leq r \leq 1, \quad 0 \leq \theta \leq 2 \pi}\end{array} $$

In Exercises \(35-44,\) use a parametrization to find the flux \(\iint_{S} \mathbf{F} \cdot \mathbf{n} d \sigma\) across the surface in the given direction. Paraboloid \(\mathbf{F}=4 x \mathbf{i}+4 y \mathbf{j}+2 \mathbf{k}\) outward (normal way from the \(z\) -axis through the surface cut from the bottom of the paraboloid \(z=x^{2}+y^{2}\) by the plane \(z=1\)

Let \(\mathbf{n}\) be the outer unit normal (normal away from the origin) of the parabolic shell $$ \text { S: } 4 x^{2}+y+z^{2}=4, \quad y \geq 0 $$ and let $$ \mathbf{F}=\left(-z+\frac{1}{2+x}\right) \mathbf{i}+\left(\tan ^{-1} y\right) \mathbf{j}+\left(x+\frac{1}{4+z}\right) \mathbf{k} $$ Find the value of $$ \iint_{S} \nabla \times \mathbf{F} \cdot \mathbf{n} d \sigma $$

Find the moment of inertia about the \(z\) -axis of a thin spherical shell \(x^{2}+y^{2}+z^{2}=a^{2}\) of constant density \(\delta .\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Calculus

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.