COVERED TODAY: The derivative matrix and differentiability. (This is not in the book.)

DUE NEXT TIME:

For each of the following functions $f$ find the derivative matrix $f^\prime$:

(1) $f(x,y)= \left( \begin{array}{c}xy x+y \end{array} \right)$

(2) $f(x,y,z) = \left( \begin{array}{c} x+\sin y y + \cos z x+y+z \end{array} \right)$

(3) $f(t,u,v) = \left( \begin{array}{c} t \cos u \cos v t \sin u \ \sin v t \cos v \end{array} \right)$

(4) $f(x) = x^2 e^x$

For the following functions find the derivative matrix at the indicated point:

(5) $f(x,y) = (e^{xy} , xy )$ at $(x,y) = (0,0)$

(6) $f(x,y,z) = xyz - x e^{yx} - y e^{xz} - ze^{xy}$ at $(x,y,z) = (1,1,1)$

(7) Let $f: R^n \longrightarrow R^m$ be a linear transformation with matrix $A$, i.e. $f(x) = A x$. What is the dervative matrix of $f$?

(8) A translation is a function of the form $T(x) = x + b$, for a fixed vector $b$. What is the derivative matrix of a translation from $R^n$ to $R^n$?

(9) Consider the function $f: R^n \longrightarrow R$ defined by $f(x) = \vert x\vert^2 = x \bullet x$. Prove that $f^\prime (x)y=2x \bullet y$, for any $x$ and $y$ in $R^n$.

(10) Verify that the function

$$f(x,y) = \left\{ \begin{array}{ll} {\displaystyle \frac{xy}{x... ... & x \not = \pm y & 0, & x = \pm y \end{array} \right.$$

has a derivative matrix at $(0,0)$, but $f$ is not differentiable there.