V2.
Gradient Fields
and Exact Differentials
1.
Criterion for gradient fields.
Let
F
=
M(x, y) i
+
N(x, y)
j
be a twodimensional vector field, where M and N are
continuous functions. There are three equivalent ways of saying that
F is conservative, i.e.,
a gradient field:
(1)
F
=
V
f
H
R
F
.
dr is pathindependent
H
F
.
dr
=
0 for any closed
C
Unfortunately, these equivalent formulations don't give us any effective way of deciding
if a given field
F is a conservative field or not. However, if we assume that F is not just
continuous but is even continuously differentiable (meaning: M,
,
My,N,
,
Ny all exist and
are continuous), then there is a simple and elegant criterion for deciding whether or not
F
is a gradient field in some region.
Criterion. Let
F
=
Mi
+
Nj be continuously differentiable in a region D. Then, in
D,
(2)
F
=
V
f for some f (x,y) My
=
N,
.
Proof. Since
F
=
Vf, this means
M
=
f,
and
N
=
fy
.
Therefore,
My=fxy
N,=fy,.
But since these two mixed partial derivatives are continuous (since My and N, are, by
hypothesis), a standard 18.02 theorem says they are equal. Thus My
=
N,.
This theorem may be expressed in a slightly different form, if we define the scalar function
called the
twodimensional curl
of
F by
(3)
curl
F
=

M,
Then (2) becomes
This criterion allows us to test
F to see if it is a gradient field. Naturally, we would also
like to know that the converse is true: if curl
F
=
0, then
F is a gradient field. As we shall
see, however, this requires some additional hypotheses on the region D. For now, we will
assume D is the whole plane. Then we have
Converse to Criterion. Let
F
=
+
Nj be continuously differentiable for all x, y.
(4)
My
=
N, for all x, y
F
=
V f
for some differentiable
f and all x, y.