Math 242, Spring 2019, Assignment 4

I was at the mathematical school, where the master taught his pupils after a method scarce imaginable to us in Europe. The proposition and demonstration were fairly written on a thin wafer, with ink composed of a cephalic tincture. This the student was to swallow upon a fasting stomach, and for three days following eat nothing but bread and water. As the wafer digested the tincture mounted to the brain, bearing the proposition along with it.

- Jonathan Swift, Gulliver's Travels

Read:

1. Section 13.1.
2. Section 13.2.
3. Section 13.3.

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

1. Vector function.
2. Components (of a vector function).
3. Domain (of a vector function).
4. Limit (of a vector function, at a point).
5. Continuous (vector function).
6. Derivtive (of a vector function).
7. Integral (of a vector function).
8. $\vec{T}(t)$ (the unit tangent vector to the space curve $\vec{r}(t)$).
9. Length (of a space curve).
10. Arc length function $s(t)$.
11. Reparametrization by arc length.
12. Curvature (of a space curve).

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

1. Theorem relating limits of vector functions to limits of their components (on page 848).
2. Theorem relating derivatives of vector functions to derivatives of their components (on page 856).
3. Differentiation rules (on page 858).
4. Theorem relating integrals of vector functions to integrals of their components (on page 859).

Solve the following problems:

1. Section 13.1, problems 1, 3, 5, 7, 21, 23, and 25.
2. Section 13.2, problems 1, 3, 9, 15, 17, 21, 23, 35, and 37.
3. Section 13.3, problems 1, 3, 13, 15, and 16.

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