Homework 4

The reading for this homework is a bit strange since we are going “off book.”  I beg your indulgences.  We will be skipping (or, rather, delaying) sections 9.6, 9.7 and 9.8 until a later time; you may read these sections if you prefer, but they will not be directly relevant to this assignment.

First, read (or re-read) pages 11-17 in our textbook.  (This is material back in section 9.1)

In preparation for the rest of this week and next week, read section 10.2 (pages 105 – 114).  You may want to first read section 10.1, which is all about limits, before reading about partial derivatives.  For this class, though, I prefer to more-or-less skip a detailed discussion of limits and jump straight to derivatives.  If you would like to get ahead of our class discussions, then you may also want to go ahead and read 10.3, too.

The Assignment

Problem 1.  Write down the equation for the plane that contains the two  lines (or explain why no plane contains these lines):

$\displaystyle \vec{\ell}(t) = \langle\, 1+2t, 2-3t, 4t \, \rangle \text{ and } x(s) = 2s, y(s) = 2+2s, z(s) = s/2.$

Note: One of the lines is given in a “vector equation” form, while there is given in “parametric” form.

Problem 2.  Consider the function

$\displaystyle f(x,y) = \sin\left(\sqrt{x^2+y^2}\right)$.

(a)  What is the domain of $f(x,y)$?

(b) Sketch a picture of the $z$-slice for $z = 1$ a.k.a. the level set

$\left\{ (x,y) : f(x,y) = 1 \right\}$.

(c ) Sketch a picture of the $x$-slice for $x = 0$, and sketch a picture of the $x$-slice for $x = 1$ (these pictures can be drawn in the $yz$ plane).

(d) Using the information from parts (a) and (b), sketch a picture of the graph $z = f(x,y)$ in $\mathbb{R}^3$.  Be sure to indicate one of the $z$-slices and one of the $x$-slices you computed before.

(e) Convert the equation $z = f(x,y)$ to one involving cylindrical coordinates.  You should, after some simplification, obtain an expression of the form $z = g(r)$ for some function $g$.  What does it mean that $\theta$ does not appear in this expression?  That is, explain how the absence of $\theta$ relates to the graph you sketched in part (d).

Problem 3.  Write down our conversion formulas for translating between rectangular, polar, cylindrical, and spherical coordinates:

$\displaystyle x = \underline{\phantom{r\cos\theta}} \text{ (polar) } \,\, r = \underline{\phantom{\sqrt{x^2+y^2}}} \text{ (rectangular) }$

$\displaystyle y = \underline{\phantom{r\sin\theta}} \text{ (polar) } \,\, \tan\theta = \underline{\phantom{y/x}} \text{ (rectangular) }$

$\displaystyle x = \underline{\phantom{r\cos\theta}} \text{ (cylindrical) } = \underline{\phantom{\rho\sin\varphi\cos\theta}} \text{ (spherical) }$

$\displaystyle y = \underline{\phantom{r\cos\theta}} \text{ (cylindrical) } = \underline{\phantom{\rho\sin\varphi\cos\theta}} \text{ (spherical) }$

$\displaystyle z = \underline{\phantom{r\cos\theta}} \text{ (cylindrical) } = \underline{\phantom{\rho\sin\varphi\cos\theta}} \text{ (spherical) }$

Problem 4.  Draw a picture of the level set

$\displaystyle \left\{ (x,y,z) \in \mathbb{R}^3 : \rho = \csc\varphi\,\sec\theta\right\}$.

Problem 5.  Compute the area of the parallelogram spanned by the two vectors $\vec{v} = \langle 1, 2, 1 \rangle$ and $\vec{w} = \langle 0, 1, 1 \rangle$.

Problem 6.  Compute the volume of the parallelpiped spanned by the three vectors $\vec{v}, \vec{w}$ and $\vec{z}$ where $\vec{v}$ and $\vec{w}$ are the same as in problem 5, and where $\vec{z} = \langle 1, 2, 3 \rangle$.

Problem 7.  Compute the partial derivatives of

$\displaystyle f(x,y) = \frac{x}{\sqrt{x^2+y^2}}$.

$f_x = \partial f/ \partial x =$

$f_y = \partial f/ \partial y =$