28.4.1 Conservative Vector Fields
Definition 28.4.2 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 28.4.3 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 points.
Conservative vector fields are important because of the following theorem, sometimes
called the fundamental theorem for line integrals.
Theorem 28.4.4 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, then
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
a polygonal curve from
. Thus the orientation is such that
goes from x0 to x. The curve p
denotes the same set of points but in the opposite
order. Suppose first
is conservative. Fix x0 ∈ U
This is well defined because if q
is another polygonal curve joining
the curve obtained by following p
and then from x
is a 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
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 28.5
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
and then from p
the above would imply the integral over the curve from x
the integral would yield ϕ
. Adding these gives
. 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 28.4.5 Let F
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 obtained,
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 28.4.6 Let F
. Find a scalar potential for F if it
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
. Therefore, this vector field is not conservative and there does not exist a scalar
Definition 28.4.7 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 sock.
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 28.4.8 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 28.4.9 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
with respect to y
, you see gy
= 1 so g
. Taking the derivative with respect to
, you get
and so it suffices to take