Math 242, Spring 2019, Assignment 14

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 16.4.
2. Section 16.5.
3. Section 16.6.
4. Section 16.7.

Carefully define the following terms, then give one example and one non-example of each:

1. Curl (of a vector field).
2. Parametric surface.
3. $\int\int_S f(x,y,z)\,dS$ (the integral of the scalar function $f$ over the parametric surface $S$).
4. Orientation (of a parametric surface).
5. Orientable (surface).
6. $\int\int_S \vec{F}\cdot\vec{n}\,dS$ (the integral of the vector field $F$ over the oriented surface $S$).
7. Flux (of a vector field across an oriented surface $S$).

Carefully state the following theorems (you do not need to prove them):

1. Clairaut test (characterizing when a vector field $\vec{F}=\left\langle P,Q\right\rangle$ is conservative, in terms of the partial derivatives of its components).
2. Green's Theorem.
3. Formula for the area element on a parametric surface.

Solve the following problems:

1. Section 16.4, problems 1, 3, 5, 7, and 28.
2. Section 16.5, problems 1(a), 3(a), and 24.
3. Section 16.6, problems 19, 23, and 25.
4. Section 16.7, problems 5, 9, 17, 21, and 25.

--------------------End of assignment--------------------