Math 242, Spring 2019, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Section 16.1.
2. Section 16.2.
3. Section 16.3.

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

1. Vector field.
2. Gradient field.
3. $\int_C\vec{F}\cdot d\vec{r}$ (the line integral of the vector field $\vec{F}$ along the curve $C$).
4. $\int_Cf(x,y)\,ds$ (the line integral of the scalar function $f$ along the curve $C$; note that we did not discuss this in class, so you will need to look in your book, on page 1075, for the definition and for a discussion of how to calculate it).
5. Conservative field.
6. $-C$ (the reversal of the curve $C$).
7. $C_1+C_2$ (the concatenation of the curve $C_2$ with the curve $C_1$).

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

1. Fundamental Theorem of Calculus for line integrals.
2. Theorem concerning path-independence of integrals of gradient fields.
3. Theorem concerning the integration of gradient fields around loops.
4. Theorem relating gradient fields to conservative fields.
5. Theorem relating line integrals along $C$ to line integrals along $-C$.
6. Theorem relating line integrals along $C_1+C_2$ to line integrals along $C_1$ and along $C_2$.

Solve the following problems:

1. Section 16.1, problems 1, 3, 5, 6, 15, 17, and 21.
2. Section 16.2, problems 1, 17, 18, 19, and 21.
3. Section 16.3, problems 1, 3, 11, 13, and 23.

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