   Chapter 16.1, Problem 26E

Chapter
Section
Textbook Problem

Find the gradient vector field ∇f of f and sketch it.26. f(x, y) = 1 2 (x2 – y2)

To determine

To find: The gradient vector field for equation f(x,y)=12(x2y2) and gradient vector field for equation f(x,y)=12(x2y2) .

Explanation

Given data:

f(x,y)=12(x2y2)

Formula used:

Write the expression for gradient vector field of two dimensional vector.

f(x,y)=fxi+fyj (1)

Consider a two-dimensional vector F=x,y .

Write the expression for length of the two dimensional vector.

|F(x,y)|=x2+y2 (2)

Write the required differentiation formulae with respect to x as follows.

x(x2)=2xx(y2)=0

Write the required differentiation formulae with respect to y as follows.

y(x2)=0y(y2)=2y

Differentiate the term 12(x2y2) with respect to x .

x(12(x2y2))=12[x(x2)x(y2)]=12[2x0]=2x2=x

Differentiate the term 12(x2y2) with respect to y .

y(12(x2y2))=12[y(x2)y(y2)]=12[02y]=2y2=y

Find the gradient vector field of f(x,y)=12(x2y2) using equation (1).

Modify equation (1) as follows.

f(x,y)=x(12(x2y2))i+y(12(x2y2))j

Substitute x for x(12(x2y2)) and y for y(12(x2y2)) ,

f(x,y)=xiyj=x,y

Thus, the gradient vector field for 12(x2y2) is xiyj_ .

Find the length of f(x,y) using equation (2).

|f(x,y)|=(x)2+(y)2=x2+y2

Consider a certain interval of x as (2,2) and y as (2,2) to plot f(x,y)

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

Solve the equations in Exercises 112 for x (mentally, if possible). x1=cx+d(c1)

Finite Mathematics and Applied Calculus (MindTap Course List)

Find the limit or show that it does not exist. limtt+t22tt2

Single Variable Calculus: Early Transcendentals, Volume I

In Exercises 7-28, perform the indicated operations and simplify each expression. 15. 4x295x26x+9

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

, where E is the wedge-shaped solid shown at the right, equals:

Study Guide for Stewart's Multivariable Calculus, 8th

An integral for the solid obtained by rotating the region at the right about the y-axis is:

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