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380 CHAPTER 12 Vector Integral Calculus

SECTION 12.3 PROBLEMS

In each of Problems 1 through 5, evaluate F · dR over −y x
C 3. F = + x 2 i + − 2y j
anysimpleclosedpathinthe x, y-plane that does not x + y 2 x + y 2
2
2
pass through the origin. This may require cases, as in −y x
Example 12.10. 4. F = + 3x i + − y j
x + y 2 x + y 2
2
2
x y
1. F = i + j x y
2
x + y 2 x + y 2 − 3y 2 j
2
5. F = + 2x i +
2
2
3/2 x + y 2 x + y 2

1
2. F = (xi + yj)
x + y 2
2
12.4 Independence of Path and Potential Theory
A vector field F is conservative if it is derivable from a potential function. This means that
for some scalar field ϕ,
∂ϕ ∂ϕ ∂ϕ
F =∇ϕ = i + j + k.
∂x ∂y ∂z
We call ϕ a potential function,or potential,for F. Of course, if ϕ is a potential, so is ϕ + c
for any constant c.

One consequence of F being conservative is that the value of F · dR depends only on
C
the endpoints of C.If C has differentiable coordinate functions x = x(t), y = y(t), z = z(t) for
a ≤ t ≤ b, then

∂ϕ ∂ϕ ∂ϕ
F · dR = dx + dy + dy
C C ∂x ∂y ∂z
b

∂ϕ dx ∂ϕ dy ∂ϕ dz
= + + dt
∂x dt ∂y dt ∂z dt
a
d
b
= ϕ(x(t), y(t), z(t))dt
a dt
= ϕ(x(b), y(b), z(b)) − ϕ(x(a), y(a), z(a))
which requires only that we evaluate the potential function for F at the endpoints of C.Thisis
the line integral version of the fundamental theorem of calculus, and applies to line integrals of
conservative vector fields.

THEOREM 12.2

Let F be conservative in a region D (of the plane or of 3-space). Let C be a path from P 0 to P 1 in
D. Then

F · dR = ϕ(P 1 ) − ϕ(P 0 ). (12.4)
C

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October 14, 2010 14:53 THM/NEIL Page-380 27410_12_ch12_p367-424