   Chapter 16.3, Problem 29E

Chapter
Section
Textbook Problem

Show that if the vector field F = P i + Q j + R k is conservative and P, Q, R have continuous first-order partial derivatives, then ∂ P ∂ y   =   ∂ Q ∂ x                   ∂ P ∂ z   =   ∂ R ∂ x                 ∂ Q ∂ z   =   ∂ R ∂ y

To determine

To show: For a conservative vector field F=Pi+Qj+Rk , there exist Py=Qx , Pz=Rx , and Qz=Ry .

Explanation

Given data:

Conservative vector field is F=Pi+Qj+Rk .

When a vector field is conservative F , then there exist a potential function such that F=f .

Consider f=fxi+fyj+fzj .

Write the relation between the potential function f and conservative vector field F .

f=F

Substitute fxi+fyj+fzj for f ,

F=fxi+fyj+fzj

Compare the equation F=fxi+fyj+fzj with F=Pi+Qj+Rk .

P=fx (1)

Q=fy (2)

R=fz (3)

Apply partial differentiation with respect to y on both sides of equation (1).

Py=y(fx)

Py=fxy (4)

Apply partial differentiation with respect to z on both sides of equation (1).

Pz=z(fx)

Pz=fxz (5)

Apply partial differentiation with respect to x on both sides of equation (2).

Qx=x(fy)=fyx

Apply Clairaut’s theorem

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Convert the expressions in Exercises 8596 radical form. 32x1/4

Finite Mathematics and Applied Calculus (MindTap Course List)

Distance Find the distance between the given numbers. 73.

Precalculus: Mathematics for Calculus (Standalone Book)

State the Second Derivatives Test.

Calculus (MindTap Course List)

Change each fraction to a percent: 56

Elementary Technical Mathematics

Evaluate the definite integral. 011+7x3dx

Single Variable Calculus: Early Transcendentals

True or False: is a convergent series.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 