13.2.3 Conservative Vector Fields
Definition 13.2.3 A vector field F defined in a three dimensional region is said
to be conservative
if for every piecewise smooth closed curve C, it follows ∫
CF⋅dR = 0.
Definition 13.2.4 Let
be an ordered list of points in ℝp.
denote the piecewise smooth curve consisting of a straight line segment from x to p1 and then
the straight line segment from p1 to p2
and finally the straight line segment from pn to y.
This is called a polygonal curve. An open set in ℝp, U, is said to be a region if it has the
property that for any two points x,y ∈ U, there exists a polygonal curve joining the two
Conservative vector fields are important because of the following theorem, sometimes called
the fundamental theorem for line integrals.
Theorem 13.2.5 Let U be a region in ℝp and let F : U → ℝp be a continuous vector
field. Then F is conservative if and only if there exists a scalar valued function of p variables
ϕ such that F = ∇ϕ. Furthermore, if C is an oriented curve which goes from x to y in U,
Thus the line integral is path independent in this case. This function ϕ is called a scalar
potential for F.
Proof: To save space and fussing over things which are unimportant, denote by p
a polygonal curve from
. Thus the orientation is such that it goes from x0
denotes the same set of points but in the opposite order. Suppose first
conservative. Fix x0 ∈ U
This is well defined because if q
is another polygonal curve joining
curve obtained by following p
and then from x
closed piecewise smooth curve and so by assumption, the line integral along this closed curve
equals 0. However, this integral is just
and that ϕ is well defined. For small t,
is open, for small t
, the ball of radius
is contained in U
Therefore, the line segment from x
is also contained in U
and so one can take
. Therefore, the above difference quotient reduces
by the mean value theorem for integrals. Here st
is some number between 0 and
1. By continuity of F,
this converges to Fi
0. Therefore, ∇ϕ
Conversely, if ∇ϕ = F, then if R :
is any C1
curve joining x
and this verifies 13.3
in the case where the curve joining the two points is smooth.
The general case follows immediately from this by using this result on each of the
pieces of the piecewise smooth curve. For example if the curve goes from x
then from p
the above would imply the integral over the curve from x
the integral would yield ϕ
. Adding these
. The formula
implies the line integral over any closed curve
equals zero because the starting and ending points of such a curve are the same.
Example 13.2.6 Let F
. Let C
be a piecewise smooth curve which goes from
. Find ∫
The specifics of the curve are not given so the problem is nonsense unless the vector field
is conservative. Therefore, it is reasonable to look for the function ϕ satisfying ∇ϕ = F. Such
a function satisfies
and so, assuming ϕ exists,
I have to add in the most general thing possible, ψ
to ensure possible solutions are not
being thrown out. It wouldn’t be good at this point to only add in a constant since the answer
could involve a function of either or both of the other variables. Now from what was just
and so it is possible to take ψy = 0. Consequently, ϕ, if it exists is of the form
Now differentiating this with respect to z gives
and this shows ψ does not depend on z either. Therefore, it suffices to take ψ = 0
Therefore, the desired line integral equals
The above process for finding ϕ will not lead you astray in the case where there does not
exist a scalar potential. As an example, consider the following.
Example 13.2.7 Let F
. Find a scalar potential for F if it exists.
If ϕ exists, then ϕx = x and so ϕ =
but this is
impossible because the left side depends only on y
while the right side depends also on
. Therefore, this vector field is not conservative and there does not exist a scalar
Definition 13.2.8 A set of points in three dimensional space V is simply
connected if every piecewise smooth closed curve C is the edge of a surface S which is
contained entirely within V in such a way that Stokes theorem holds for the surface S
and its edge, C.
This is like a sock. The surface is the sock and the curve C goes around the opening of the
As an application of Stoke’s theorem, here is a useful theorem which gives a way to check
whether a vector field is conservative.
Theorem 13.2.9 For a three dimensional simply connected open set V and F
a C1 vector field defined in V , F is conservative if ∇× F = 0 in V .
Proof: If ∇× F = 0 then taking an arbitrary closed curve C, and letting S be a surface
bounded by C which is contained in V , Stoke’s theorem implies
Thus F is conservative. ■
Example 13.2.10 Determine whether the vector field
Since this vector field is defined on all of ℝ3, it only remains to take its curl and see if it is
the zero vector.
This is obviously equal to zero. Therefore, the given vector field is conservative.
Can you find a potential function for it? Let ϕ be the potential function. Then
ϕz = 2
and so ϕ
. Now taking
the derivative of
with respect to y
, you see gy
= 1 so g
. Taking the derivative with respect to
, you get
and so it suffices to take