26. Stokes' Theorem

Let \(S\) be a nice surface in \(\mathbf{R}^3\) with a nice properly oriented boundary, \(\partial S\), and let \(\vec{F}\) be a nice vector field on \(S\). Then \[ \iint_S \vec{\nabla}\times\vec{F}\cdot d\vec{S} =\oint_{\partial S} \vec{F}\cdot d\vec{s} \] as seen from the tip of the normal vector to the surface.

Homework

1. Verify Stoke's Theorem on the frustum of the cone \(z=r\) for \(2 \le z \le 4\), oriented down and out for the vector field \(\vec G=\langle -yz^2,xz^2,z(x^2+y^2)\rangle\).

   

   1. Compute the curl in rectangular coordinates. Parametrize the surface and evaluate the curl on the surface. Find the tangent and normal vectors and check the orientation. Then compute the integral.

      Be sure to compute the tangent and normal vectors starting from the parametrization, \(\vec R\), not from the curl of \(\vec F\).

   2. Parameterize each circle. Compute the tangent vectors and check the orientations. Evaluate \(\vec{F}\) on each circle and compute the circulation around each circle. Finally add the line integrals and compare the answer with part (a).

2. Compute \(\displaystyle \int_C \vec F\cdot d\vec s\) for \(\vec F=\langle-yx^2,xy^2,z^3\rangle\) over the curve \(z=xy\) drawn on the cylinder \(x^2+y^2=4\) traversed countercockwise as seen from the positive \(z\)-axis.

   

   Notice that this curve is the boundary of the surface \(z=xy\) within the cylinder \(x^2+y^2=4\).

   

3. Compute \(\displaystyle \iint_S \vec{\nabla}\times\vec{F}\cdot d\vec{S}\) for the vector field \(\vec{F}=\langle-yz^2,xz^2,z\rangle\) over the surface \(S\) given by \(z=\cos^2(2\pi\sqrt{x^2+y^2})\) for \(x^2+y^2 \le 9\), oriented upward.
   Note: The surface may be parametrized by: \[ \vec R=\langle r\cos\theta,r\sin\theta,\cos^2(2\pi r)\rangle \]