14.3 The Fundamental Theorem for Line Integrals
Conservative Vector Field
Denition. A vector eld F is called a conservative vector eld if it is the gradient
of some scalar function f , that is, F = ∇f . In this situation f is called a potential
function for F.
Example 1. Gravitational Field: Suppose an object with mass M is located at the
origin in R3 . By Newton's Law of Gravitation, the gravitational force acting on the
object with mass m at x = hx, y, zi is
F(x) = −
mM G
mM Gx
mM Gy
mM Gz
x
=
h−
,
−
,
−
i
|x|3
(x2 + y 2 + z 2 )3/2 (x2 + y 2 + z 2 )3/2 (x2 + y 2 + z 2 )3/2
where G is the gravitational constant.
The gravitational eld F is conservative because if we dene
mM G
f (x, y, z) = p
x2 + y 2 + z 2
then ∇f = F.
The Fundamental Theorem for Line Integrals
Recall that the Fundamental Theorem of Calculus can be written as
ˆ
b
f 0 (x) dx = f (b) − f (a)
a
Theorem. Let C be a smooth curve given by the vector function r(t), a ≤ t ≤ b.
Let f be a dierentiable function of two or three variables whose gradient vector ∇f
is continuous on C . Then
ˆ
∇f · dr = f (r(b)) − f (r(a))
C
Example 2. Find the work done by the gravitational eld
F(x, y, z) = h−
my
mz
mx
,
−
,
−
i
(x2 + y 2 + z 2 )3/2 (x2 + y 2 + z 2 )3/2 (x2 + y 2 + z 2 )3/2
in moving a particle with mass m from the point (1, 2, 2) to the point (3, 4, 12) along
a piecewise-smooth curve C .
Independence of path
Suppose C1 and C2 are two piecewise-smooth curves (which are
´ called paths
´ ) that
have the same initial point A and terminal point B . In general, C1 F · dr 6= C2 F · dr
for any vector eld F.
But, according the Fundamental
´
´ Theorem, if F is conservative, i.e., F = ∇f for
some function f , then C1 F · dr = C2 F · dr. In fact, the Fundamental Theorem says
that the line integral of ∇f is the net change in f that only depends on initial value
f (A) and terminal value f (B).
In general, if F is a continuous´ vector eld´with domain D, we say that the line integral
is independent of path if C1 F · dr = C2 F · dr for any two paths C1 and C2 in D
that have the same initial and terminal points.
Theorem.
´
C F · dr is independent of path in D if and only if
closed path in D.
´
C
F · dr = 0 for every
As a consequence, the work done by a conservative force eld (such as the gravitational
or electric eld) as it moves an object around a closed path is 0.
The following theorem says that the only vector elds that are independent of path
are conservative.
Theorem
´ . Suppose F is a vector eld that is continuous on an open connected region
D. If C F · dr is independent of path in D, then F is a conservative vector eld on
D; that is, there exists a function f such that F = ∇f .
Conservative Vector Field on Plane
Theorem. If F(x, y) = P (x, y)i + Q(x, y)j is a conservative vector elds, where P
and Q have continuous rst-order partial derivatives on a domain D, then throughout
D we have
∂P
∂Q
=
∂y
∂x
The converse of Theorem is true only for a special type of the region.
Denition. A simply-connected region in the plane is a connected region D which
is connected and contains no hole.
Theorem. If F(x, y) = P (x, y)i + Q(x, y)j is a vector eld on an open simplyconnected region D. Suppose that P and Q have continuous rst-order derivatives
and
∂P
∂Q
=
∂y
∂x
Then F is conservative.
Example 3. Determine whether or not the vector eld is conservative:
(a) F(x, y) = hx2 + y 2 , 2xyi
(b) F(x, y) = hx2 + 3y 2 + 2, 3x + yey i
Example 4. Given F(x, y) = sin yi + (x cos y + sin y)j
(a) Show that F is conservative.
(b) Find a function f such that ∇f = F.
(c) Find the work done by the force eld F in moving a particle from the point (3, 0)
to the point (0, π/2).