   Chapter 16.1, Problem 24E

Chapter
Section
Textbook Problem

Find the gradient vector field of f.24. f(x, y, z) = x2 yey/z

To determine

To find: The gradient vector field for equation f(x,y,z)=x2yeyz .

Explanation

Given data:

f(x,y,z)=x2yeyz .

Formula used:

Write the expression for gradient vector field of a three-dimensional vector.

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

Write the required differentiation formulae as follows.

t[u(t)v(t)]=u(t)t[v(t)]+v(t)t[u(t)] (2)

x(x2)=2xx(ex)=exy(eyz)=eyz(1z)z(eyz)=eyz(yz2)

Differentiate the term x2yeyz with respect to x .

x(x2yeyz)=yeyzx(x2)=yeyz(2x)=2xyeyz

Differentiate the term x2yeyz with respect to y .

y(x2yeyz)=x2y(yeyz)=x2[yy(eyz)+eyzy(y)]=x2[yeyz(1z)+eyz(1)]=x2eyz[yz+1]

Differentiate the term x2yeyz with respect to z

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