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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

Let r = x i + y j + z k and r = |r|.

31. Verify each identity.

(a) ∇r = r/r

(b) ∇ × r = 0

(c) ∇(1/r) = −r/r3

(d) ∇ ln r = r/r2

(a)

To determine

To verify: The vector field of the form r=rr .

Explanation

Given data:

r=xi+yj+zk

Find r .

r=(xi+yj+zk)(|xi+yj+zk|)=(xi+yj+zk)(x2+y2+z2)=x(x2+y2+z2)i+yy(x2+y2+z2)j+zz(x2+y2+z2)k=[12(x2+y2+z2)12

(b)

To determine

To verify: The vector field of the form ×r=0 .

(c)

To determine

To verify: The vector field of the form (1r)=rr3 .

(d)

To determine

To verify: The vector field of the form lnr=rr2 .

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Chapter 16 Solutions

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