Calculus 2nd Edition Rogawski

SECTION 17.1 Vector Fields 949
THEOREM 2 Uniqueness of Potential Functions If F is conservative on an open
connected domain, then any two potential functions of F differ by a constant.
Proof If both V 1 and V 2 are potential functions of F, then
∇(V 1 − V 2 ) = ∇V 1 −∇V 2 = F − F = 0
However, a function whose gradient is zero on an open connected domain is a constant
function (this generalizes the fact from single-variable calculus that a function on an
interval with zero derivative is a constant function—see Exercise 36). Thus V 1 − V 2 = C
for some constant C, and hence V 1 = V 2 + C.
The next two examples consider two important radial vector fields.
The result of Example 5 is valid in R 2 : The
function
V (x, y) = √ x 2 + y 2 = r
is a potential function for e r .
where
In R 2 ,
REMINDER
〈 x
e r =
r , y r , z 〉
r
r = (x 2 + y 2 + z 2 ) 1/2
〈 x
e r =
r , y 〉
r
where r = (x 2 + y 2 ) 1/2 .
z
EXAMPLE 5 Unit Radial Vector Fields Show that
√
V (x, y, z) = r = x 2 + y 2 + z 2
is a potential function for e r . That is, e r = ∇r.
Solution We have
Similarly, ∂r
∂y = y r
∂r
∂x = ∂
∂x
√
x 2 + y 2 + z 2 =
x
√
x 2 + y 2 + z 2 = x r
∂r
and
∂z = z 〈 x
r . Therefore, ∇r = r , y r , z 〉
= e r .
r
The gravitational force exerted by a point mass m is described by an inverse-square
force field (Figure 9). A point mass located at the origin exerts a gravitational force F on
a unit mass located at (x, y, z) equal to
F = − Gm
r 2 e r = −Gm
〈 x
r 3 , y r 3 , z
r 3 〉
where G is the universal gravitation constant. The minus sign indicates that the force is
attractive (it pulls in the direction of the origin). The electrostatic force field due to a
charged particle is also an inverse-square vector field. The next example shows that these
vector fields are conservative.
EXAMPLE 6 Inverse-Square Vector Field Show that
( )
e r −1
r 2 = ∇ r
Solution Use the Chain Rule for gradients (Theorem 1 in Section 15.5) and Example 5:
∇(−r −1 ) = r −2 ∇r = r −2 e r
x
y
17.1 SUMMARY
FIGURE 9 The vector field − Gme r
r 2
represents the force of gravitational
attraction due to a point mass located at the
origin.
• Avector field assigns a vector to each point in a domain.Avector field in R 3 is represented
by a triple of functions
F = ⟨F 1 ,F 2 ,F 3 ⟩
A vector field in R 2 is represented by a pair of functions F = ⟨F 1 ,F 2 ⟩. We always assume
that the components F j are smooth functions on their domains.