# Vector HW 7

Problem 1.  Verify that the Chain Rule (as stated in class) is true for the functions

$\displaystyle F(x,y) = (x+y, ye^x) \text{ and } G(s, t) = (s+t, s-t, s^2)$.

Do this by first writing out their composition as a function $H$ and then computing the derivative matrix $DH$.

Next, compute the derivative matrices $DF$ and $DG$ and multiply them.  Your final answer should equal $DH$.

Problem 2.  Consider the function $f(x,y) = y/x$.  Use the Chain Rule to compute $\partial f/ \partial \theta$ where $\theta$ is our standard cylindrical coordinate variable.

Problem 3.  (a) Suppose $G: \mathbb{R}^3 \to \mathbb{R}^3$ is a differentiable function and that at the point $\vec{x}_0 = \left(1, 2, 1\right)$ we know that $G(\vec{x}_0) = \left(0, 0, 2\right)$ and we know that

$\displaystyle DG\big{|}_{\vec{x}_0} = \left[ \begin{array}{ccc} 1 & 2 & \pi \\ 0 & 1 & -1 \\ -1 & 7 & 1/2 \end{array} \right]$

Compute the linear approximation, $L(x,y,z)$, for $G$ at the point $\vec{x}_0$.

(b) Suppose $F:\mathbb{R}^3 \to \mathbb{R}^2$ is a differentiable function and that at the point $\vec{z}_0 = \left(0, 0, 2\right)$ we know that

$\displaystyle DF \big{|}_{\vec{z}_0} = \left[ \begin{array}{ccc} 1 & 1 & 1 \\ \sqrt{2} & -2 & 5 \end{array}\right]$.

Compute the derivative matrix $D\left(F\circ G\right)$ at the point $\vec{x}_0 = (1, 2, 1)$ where $G$ is the function from part (a).

Problem 4. Find the equation of the plane that passes through the point $\vec{x}_0 = (1, 2, 4)$ and is parallel to the plane $x - 4y + 2 = 0$.

Problem 5.  Draw a picture of the set $S = \left\{(x,y,z) \in \mathbb{R}^3 : x^2+4x + y^2 + z^2 = -3 \right\}$.

Problem 6.  Draw a picture of the line parameterized by

$\displaystyle \ell(t) = (1+t, 1+t)$.

Problem 7.  Consider the function $f(x,y) = x^3(x^2-y^2)$.

(a) Compute the derivative matrix $Df$.

(b) Compute the gradient $\nabla f$.

(c) Form the new function $g:\mathbb{R}\to\mathbb{R}$ defined by

$\displaystyle g(t) = \left( f\circ \ell\right)\,(t) = f\left(\ell(t)\right)$

where $\ell(t)$ is the function that parameterized the line in problem 6.  Note that this function $g(t)$ is the kind of function we might see in a Calculus I class; for old time’s sake, compute the derivative $g'(t)$.

(d) Use the Chain Rule to compute the $1\times 1$ derivative matrix

$\displaystyle Dg$

where $g(t)$ is the same composed function from part (c).

Note:  Your computation for part (d) could have been re-phrased in terms of the gradient of $f$ instead of its derivative matrix, $Df$.  Had we done this, we would have found

$\displaystyle g'(t) = \frac{d}{dt}\, \left(f\circ \ell\right) = \nabla f \cdot \ell'(t)$.

This is an important note!  It is our definition of something called a directional derivative!  That is, we think of this computation as telling us how $f$ changes at the point $\ell(0) = (1, 1)$ when we move inputs in the direction of $\ell$!