We now generalise Green’s theorem to vector fields in The following result is by Sir George Gabriel Stokes.
Let be a simple closed positively oriented curve in the plane parameterised by a continuously differentiable function and let be the region enclosed by Let be a twice continuously differentiable and one-to-one parameterisation of a surface . The boundary of is parameterised by the positively oriented curve
Notice that the orientation of the surface and boundary curve is specified in the theorem above. This orientation of the surface and curve can also be described in the following way: If you walk in the positive direction along the boundary curve with your head pointing in the direction of the normal vector , then the surface will always be on your left.
Use Stokes’ theorem to evaluate the integral where and is the part of the sphere that lies inside the cylinder and above the -plane.
Prove that if everywhere on , then is conservative.
We have previously seen that Green’s theorem does not apply to the case where the curve is the circle centered at with radius and where since is not defined at
Consider now the vector field
Notice that for the first two components of are the same as the first two components of Hence where is the circle centered at with radius in the -plane. Let the surface be the upper hemisphere, i.e. and Verify Stokes’ theorem for this case.
Let the curve be the boundary curve of the Möbius strip parameterised by
where Calculate the line integral
Note that the integrand is conservative. Further recall that the Möbius strip is not orientable and hence the surface integral over the Möbius strip is not defined.
Proof of Stokes’ theorem
Let the vector field Let the parameterisation of the surface be given by
Then the boundary is parameterised by
We can now write the line integral as
Using the chain rule we have
To shorten the expressions we assume in the following that and that and similarly for Thus
where the line integral is over the boundary curve of in positive orientation.
Since the last 3 integrals are over a region in and the boundary curve of is positively oriented, we can use Green’s theorem to obtain
We now apply the chain rule to the partial derivatives appearing above to obtain
By simplifying also the other expressions in the same manner we obtain