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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 $X = (u^1)\partial_1 +\partial_2$

user130306
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  • Have you computed $\phi_u$ and $\phi_v$, and the dot products of these with each other and themselves? If so, show your work. If not, why not? – John Hughes May 21 '18 at 02:21
  • On a related note: what text are you using? That'll help us be consistent on notation. – John Hughes May 21 '18 at 02:22
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    It has been viewed $5$ times and has $5$ upvotes, including my own. Wow! Very good question!! $\color{orange}{\bigstar}$ – Mr Pie May 21 '18 at 02:24
  • Sorry, I will be updating with my work for part a, I seem to be understanding a bit more – user130306 May 21 '18 at 02:33
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    Wow...I differ. I was going to downvote, but I seldom do that. What's wrong with it? There's no indication that the OP has bothered to learn the definition of "induced metric", for instance, or has expended any effort beyond formatting the LaTeX nicely. It also appears that OP doesn't know how a metric operates on a vector, since question "b" doesn't require anything more than the metric from question "a" (and an awareness of the basis in which that metric is expressed as a quadratic form). So I'm feeling like "OP copied a standard question from a diff'l geometry book. Why all the love?" – John Hughes May 21 '18 at 02:36
  • I just updated with my work – user130306 May 21 '18 at 02:41
  • And what values did you get for the $g_{ij}$? For part $b$, do you know what $\partial_1$ denotes? Without that, you can't make a lot of sense of the question. If you DO know, go ahead and tell us. – John Hughes May 21 '18 at 02:48
  • I just added my values for $g_{ij}$ and for part b, I am confused about this part: I know the notation if in the form $X = \alpha \phi_u+\beta\phi_v$ and $g_{ij}$ but I don't know now what the $\partial_1$ denotes – user130306 May 21 '18 at 03:02
  • I have also tried to solve c, I will update shortly – user130306 May 21 '18 at 03:40

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