Calculus: Early Transcendentals (2nd Edition)

2nd Edition

ISBN: 9780321947345

Author: William L. Briggs, Lyle Cochran, Bernard Gillett

Publisher: PEARSON

See similar textbooks

Concept explainers

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic…

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along…

Triple Integral

Examples:

Videos

Textbook Question

Chapter 14.7, Problem 42E

Radial fields and zero circulation Consider the radial vector fields F = r/|r|^p, where p is a real number and r = 〈x, y, z〉. Let C be any circle in the xy-plane centered at the origin.

a. Evaluate a line integral to show that the field has zero circulation on C.

b. For what values of p does Stokes’ Theorem apply? For those values of p, use the surface integral in Stokes’ Theorem to show that the field has zero circulation on C.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.7, Problem 41E

Chapter 14.7, Problem 43E

Students have asked these similar questions

■ Use Stokes’ Theorem to evaluate curl F dS

Zero curl Consider the vector field y F i + sj+ zk. x? + y x? + y? a. Show that V × F = 0. b. Show that fF · dr is not zero on a circle C in the xy-plane enclosing the origin. c. Explain why Stokes' Theorem does not apply in this case.

Stokes's Theorem / The Curl Theorem 2. Consider the vector field F(x, y, 2) = (yz?,x,x + y) and the closed curve C: r(t) = (cos t, sin t,0) 0sts 27 Note that C is the unit circle in the xy-plane traced out counterclockwise (as viewed from above). Also let D be the unit disc in the xy-plane and let E be the 45-degree cone whose tip is at the point (-1,0,0) and whose boundary is the curve C. That is, D= { (x,y,2) | x² + y²<1 and z = 0 } E = { (x,y.2)| z = J + y? – 1 and – 1szs0} a) Graph C,D, and E so that we can see what's going on. Note that D would be input as just z = 0, cut off by C as its boundary. You should note that E lines up with C as its boundary also. b) Find V x F, the curl of F. You will use this below. Now our goal is to verify the Curl Theorem, and again, we'll do it twice. The Curl Theorem claims that fF- dr = |v x F) - as = v×F) - as c) First evaluate the leftmost expression directly, the line integral of F along the closed curve C. d) Next evaluate the middle…

Chapter 14 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 14.1 - How is a vector field F = f, g, h used to describe...Ch. 14.1 - Sketch the vector field F = x, y.Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Interpret the gradient field of the temperature...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...Ch. 14.1 - Two-dimensional vector fields Sketch the following...

Additional Math Textbook Solutions

Find more solutions based on key concepts

Use the Substitution Formula in Theorem 7 to evaluate the integrals in Exercises 1–48. 1.

Thomas' Calculus: Early Transcendentals (14th Edition)

The solution of the equation tan2θtanθ=1 on the interval 0,2π .

Precalculus

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equation. x = 7

University Calculus: Early Transcendentals (3rd Edition)

In Problems 31-42: (a) Find the domain of each function. (b) Locate any intercepts. (c) Graph each function. (d...

Precalculus

Sine substitution Evaluate the following integrals. 7. 05/2dx25x2

Single Variable Calculus: Early Transcendentals (2nd Edition) - Standalone book

In Problems 25-32, the graph of a function is given. Use the graph to find: a. The intercepts, if any b. The do...

Precalculus (10th Edition)

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Solution Summary: The author evaluates the line integral to show that the field has zero circulation on C. Since the curve is in counterclockwise orientation, the normal vector of S head outwards.

Concept explainers

Videos

Want to see the full answer?

Chapter 14 Solutions

Additional Math Textbook Solutions