Activity 13.13.4

Because Stokes' Theorem requires us to consider a surface (with normal vector) and the boundary of the surface, this activity will give you a chance to practice identifying the boundary of some surfaces in $\R^3$. For each surface below:

1. Describe the boundary in words.
2. Find a parametrization for the boundary.
3. Ensure that a person walking along the boundary in the direction of your parametrization with head pointing in the direction of the surface's normal vector would hold their left hand over the surface.

1. The surface $S_1$ is the portion of the sphere $x^2+y^2+z^2=4$ with $z\geq x$. Assume the outward orientation on the sphere.

2. The surface $S_2$ is the portion of the sphere $x^2+y^2+z^2=4$ with $z\geq 0$. Assume the outward orientation on the sphere.

3. The surface $S_3$ is the portion of the hyperbolic paraboloid $z=x^2-y^2$ with $x^2+y^2\leq 1$. Assume the “upward” orientation, e.g., the normal vector at $(0,0,0)$ is $\vk$.

4. The surface $S_4$ is the portion of the cylinder $x^2+y^2=4$ for which $-2\leq z\leq 2$, assuming the outward orientation.