Solved: Investigation (a) Use a computer algebra system to graph the vector-valued

Chapter 15, Problem 43

(choose chapter or problem)

Investigation

(a) Use a computer algebra system to graph the vector-valued function

\(\mathbf{r}(u, v)=(4-v \sin u) \cos (2 u) \mathbf{i}+(4-v \sin u) \sin (2 u) \mathbf{j}+\)

\(v \cos u \mathbf{k}, \quad 0 \leq u \leq \pi, \quad-1 \leq v \leq 1\)

This surface is called a Möbius strip.

(b) Is the surface orientable? Explain why or why not.

(c) Use a computer algebra system to graph the space curve represented by r(u, 0). Identify the curve.

(d) Construct a Möbius strip by cutting a strip of paper, making a single twist, and pasting the ends together.

(e) Cut the Möbius strip along the space curve graphed in part (c), and describe the result.

Text Transcription:

r(u, v) = (4 - v sin u) cos (2u)i + (4 - v sin u) sin (2u)j +

v cos uk,     0 leq u leq pi,     -1 leq v leq 1