Page 402 - Advanced engineering mathematics

382 CHAPTER 12 Vector Integral Calculus

Conversely, suppose F · dR is independent of path in D and let C be any closed path in
C
D. Choose distinct points P 0 and P 1 on C.Let C 1 be the part of C from P 0 to P 1 and C 2 the part
C 2 . Furthermore, C 1 and −C 2 are paths in D from P 0 to P 1 ,soby

from P 1 to P 0 . Then C = C 1
assumption

F · dR = F · dR =− F · dR.
C 1 −C 2 C 2
Then

0 = F · dR + F · dR = F · dR.
C 1 C 2 C

Thus far, we have the following implications for F · dR over paths in some region D:
C

1. Conservative F ⇒ independence of path of F · dR.
C
2. Independence of path in D ⇐⇒ integrals over all closed paths in D are zero.
We will improve on this table of implications shortly. First, consider the problem of finding
a potential function for a conservative vector field. Sometimes this can be done by integration.

EXAMPLE 12.11

We will determine if the vector field
2
3 2
z
3
2
z
F(x, y, z) = 3x yz i + (x z + e )j + (2x yz + ye )k.
is conservative by attempting to find a potential function.
If F =∇ϕ for some ϕ, then
∂ϕ
2
2
= 3x yz , (12.5)
∂x
∂ϕ
z
3 2
= x z + e , (12.6)
∂y
∂ϕ 3 z
= 2x yz + ye . (12.7)
∂z
Choose one of these equations, say 12.5. To reverse ∂ϕ/∂x, integrate this equation with respect
to x to get

2
2
2
3
ϕ(x, y, z) = 3x yz dx = x yz + α(y, z).
The “constant of integration” may involve y and z because the integration reverses a partial
differentiation in which y and z were held fixed. Now we know ϕ to within α(y, z). To determine
α(x, y), choose one of the other equations, say 12.6, to get
∂ϕ ∂
3 2
3
z
2
= x z + e = (x yz + α(y, z))
∂y ∂y
∂α(y, z)
3 2
= x z + .
∂y
This requires that
∂α(y, z) z
= e .
∂y
Integrate this with respect to y, holding z fixed to get
∂α(y, z)

z
dy = ye + β(z),
∂y
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
October 14, 2010 14:53 THM/NEIL Page-382 27410_12_ch12_p367-424