Stokes' Theorem

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 hea…

In some sense, this activity considers the reverse problem of that considered in this activity. Here, each part of the activity gives you an oriented simple closed curve $C$ in $\R^3$, and your task is to find

• a surface $S$ so that $C$ is the boundary of $S$ and
• a normal vector for the $S$ so that a person walking along $C$ in the direction of the given orientation with head pointing in the direction of your chosen…

Stokes' Theorem is generally used to turn a line integral into a flux integral. Sometimes it is possible to be given a flux integral and recognize that the given vector field $\vF$ is $\curl(\vG)$ for some vector field $\vG$, however. When this is the case, we call $\vG$ a vector potential for the vector field $\vF$, much like a function $f$ so that $\vF = \grad(f)$ is called a potential function for $\vF$.

1. Find a vector field $\vF$ so that $\curl(\vF) = \langle x-y^2,4xy-y-4,-4xz\rangle$.

2. When f…