I'm unclear on how to solve any of these problems, can someone please help me with what to do?
a) Let S be the upper portion of a cone in $\mathbb{R}^3$ parametrized by $\phi(u^1,u^2) = (u^1\cos u^2, u^1\sin u^2)$; $u^1 >0$. Show that the induced metric on S from $\phi$ is
$$\begin{pmatrix} 2 & 0 \\\ 0 & (u^1)^2\end{pmatrix}$$
$\phi u_1 = <\cos u^2, \sin u^2, 1>$ and
$\phi u_2 = <-u^1\sin u^2, u^1\cos u^2, 0>$
Now do all I have to do is dot product these together?
So, $g_{11} = \phi u_1 \cdot \phi u_1 = \cos^2 u^2 + \sin^2 u^2 +1$
$g_{12} = g_{21} = \phi u_1 \cdot \phi u_2 = -u^1\sin u^2 \cos u^2+u^1\cos u^2 \sin u^2$
$g_{22} = \phi u_2 \cdot \phi u_2 = (u^1)^2\sin^2 u^2 +(u^1)^2 \cos u^2$
b) Let $X = (u^1)\partial_1 +\partial_2$, be a vector field on S. Find the length of the vector field X with respect to the metric g.
c) Suppose $f: S \to \mathbb{R}$ by $f(u^1\cos u^2, u^1\sin u^2, u^1) = (u^1)^2 -(u^1)^2-(u^1)(u^2)$Find X(f), where $X = (u^1)\partial_1 +\partial_2$