Write Both integrals involve terms and terms and terms. We shall show that the terms in the two integrals agree. In other words, we shall assume that The proofs that the and terms also agree are similar. For simplicity, we’ll assume that the boundary of consists of just a single curve, and that we can
Then the curve bounding can be parametrized as
The orientation of
We’ll now verify that the direction of increasing for the parametrization of is the direction of the arrow on in the figure on the left above.
By continuity, it suffices to check the orientation at a single point.
Find a point on where the forward pointing tangent vector is a positive multiple of The horizontal arrow on in the figure on the left below is at such a point. Suppose that at this point — in other words, suppose that Because the forward pointing tangent vector to at namely is a positive multiple of we have and The tangent vector to at pointing in the direction of increasing is
and so is a positive multiple of See the figure on the right below.
If we now walk along a path in the -plane which starts at holds fixed at and increases we move into the interior of starting at Correspondingly, if we walk along the path, in with starting at and increasing, we move into the interior of The forward tangent to this new path, points from into the interior of It’s the blue arrow in the figure on the right below.
Now imagine that you are walking along in the direction of increasing At time you are at You point your right arm straight ahead of you. So it is pointing in the direction You point your left arm out sideways into the interior of It is pointing in the direction If the direction of increasing is the same as the forward direction of the orientation of then the vector from our feet to our head, which is should be pointing in the same direction as And since it is.
Now, with our parametrization and orientation sorted out, we can examine the integrals.
The surface integral:
Since so that
and
and
Now we examine the line integral and show that it equals this one.
The line integral:
We can write this as the line integral
around if we choose
Finally, we show that the surface integral equals the line integral:
By Green’s Theorem, we have
which is the conclusion that we wanted.