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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Find the gradient vector field of f.

23. f(x, y, z) = x 2 + y 2 + z 2

To determine

To find: The gradient vector field for equation f(x,y,z)=x2+y2+z2 .

Explanation

Given data:

f(x,y,z)=x2+y2+z2

Formula used:

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

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

Write the required differentiation formulae as follows.

x[x]=12xx(x2)=2x

Differentiate terms with respect to x .

xx2+y2+z2=12x2+y2+z2x(x2)=12x2+y2+z22x=xx2+y2+z2

Differentiate terms with respect to y .

yx2+y2+z2=12x2+y2+z2y(y2)=12x2+y2+z22y=yx2+y2+z2

Differentiate terms with respect to z .

zx2+y2+z2=12x2+y2+z2z(z2)=12x2+y2+z22z=zx2+y2+z2

Find the gradient vector field of f(x,y,z)=x2+y2+z2 using equation (1)

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Chapter 16 Solutions

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