Additional exercises 1

Exercise

Let $f(x)=(x,x^2,x^3)$ and $g(x,y,z)=x+y-z$. Identify the domain and codomain of $f$, $g$. Determine if $f\circ g$ and $g\circ f$ are well defined. If it is well defined then write the function explicitly and determine the domain and codomain.

Exercise

Let $f(x,y)=x^2+5xy+\ln y$. Compute the partial derivatives $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial y}$.

Exercise

Let $f(x,y)=x^3+5+e^{xy}$. Calculate $\mathrm{\nabla }f$, the gradient of $f$.

Exercise

Determine the directional derivative, $D_{v}f(-1,4,6)$ for $f(x,y,z)=e^{x{y}^2}+4z{y}^3$ in the direction of $v=(2,-3,6)$.

Exercise

Compute the Jacobian matrix of the transformation

$$(u,v)\mapsto (e^u\cos v,e^u\sin v).$$

Exercise

Consider the functions,

$$\mathit{\alpha }:[0,\mathrm{\infty })\to {\mathbb{R}}^2;t\mapsto (t\cos t,t\sin t),$$$$f:{\mathbb{R}}^2\to \mathbb{R};(x,y)\mapsto 3x^2+y^2.$$

Let $g=f\circ \mathit{\alpha }$. Calculate $g^{\prime }$ both by using the chain rule and by first calculating $g$ and then differentiating and confirm that the answer is the same using either method. Out of curiosity, try to understand what curve $\alpha (t)$ traces out as $t$ varies.

Exercise

Let $f$ be defined, for $(x,y)\ne (0,0)$, as

$$f(x,y)=\frac{xy}{x^2+y^2},$$

and $f(x,y)=0$. Calculate the 4 missing values in the following table, assuming that $t\ne 0$.

line	value
$\{x=0\}$	$f(0,t)=$ ?
$\{y=0\}$	$f(t,0)=$ ?
$\{x=y\}$	$f(t,t)=$ ?
$\{x=-y\}$	$f(t,-t)=$ ?

What does this say about $f$ when approaching $(0,0)$ along these different lines?

Exercise

Consider the surface $x^2+y^2-z^2=1$. Verify that the point $(1,1,1)$ is contained in the surface. Find the tangent plane to this surface at this point.

Hint: write this surface as a level set $\{(x,y,z):f(x,y,z)=c\}$, calculate $\mathrm{\nabla }f$ at the specified point and use the known connection between gradient and tangent plane.