Let
be a smooth (differentiable) two-component vector field (or the pair of functions
) on the two dimensional space then the line integral of the field projection onto the unite length vector anti-clockwise field
always smoothly tangent to the closed curved over the arbitrary two dimensional closed curve
equals the integral of the difference of the partial derivatives
, and
over the plane region
bounded inside the curve or otherwise the outside of the curve values of the field make virtually no contributions to the integral over the region providing that the field is sufficiently smooth that the second derivatives of the field components exists in the region i.e.
where
and
is the region enclosed by the curve
.
We can approximate the integral on the right side over the region by the finite sum by dividing densely the space around the region
into small squares with the sides
and the vertices
and approximating the bounding curve
of the region by the sides of squares which are the closet to the curve as well as the coordinate derivatives
of the field
by their difference quotients. We will keep the vertices coordinate names for the convenience even if they are equal and keep the square vertices coordinate indices
even if they are limited by the region bounded by the curve.
We get
Now the essential in proving the theorem is to focus on the contribution to the finite sum approximating the region integral from the one component of the
field itself and notice that because of the cancelation of the sign alternating term the sums reduce to only the end points. For example for
and the fixed
-line
and its length we have
,
Note that while
is an infinitesimal (small) linear element of the region boundary curve parallel to the
axis and for the unite vector
parallel to it
and so for the second point with the minus sign the right side is an approximate to the growth
of the counter-clockwise line integral
i.e.
.
Summing up all the all the
contributions over
and repeating the considerations for the field component
leading
to the
contributions of the region boundary curve integral we get
and so finally prove
.