Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

A Find the determinant of the metric on some graph

  1. Feb 27, 2017 #1
    Hello there,

    Suppose $f$ smoothly maps a domain ##U## of ##\mathbb{R}^2## into ##\mathbb{R}^3## by the formula ##f(x,y) = (x,y,F(x,y))##. We know that ##M = f(U)## is a smooth manifold if ##U## is open in ##\mathbb{R}^2##. Now I want to find the determinant of the metric in order to compute the area of the manifold
    $$
    I = \int 1 |g|^{1/2} d^2x
    $$
    I guess that the metric on ##\mathbb{R}^n## is the Kronecker delta, so that
    $$
    g_{ij} = \frac{d\xi^a}{dx^i} \frac{d\xi^b}{dx^j} \delta_{ab}
    $$
    So if I can find ##\xi^a##, my task is easy. How do I determine ##\xi^a##. Any hints/help/solutions? Thanks
     
  2. jcsd
  3. Feb 28, 2017 #2
    Maybe I'm wrong, but I think your ##\xi^\rho## are given by the transformation law between the Cartesian and your new coordinate system. Maybe some mentor can correct me if I'm mistaken it.
     
  4. Feb 28, 2017 #3

    PeterDonis

    User Avatar
    2016 Award

    Staff: Mentor

    The metric of what?

    What do ##\xi^a## represent?
     
  5. Mar 1, 2017 #4

    Ben Niehoff

    User Avatar
    Science Advisor
    Gold Member

    This formula gives the "pullback" of the Euclidean metric on ##\mathbb{R}^3## (##\delta_{ab}## where ##a,b \in \{1,2,3\}##) to the metric on your embedded surface (##g_{ij}## where ##i,j \in \{1,2\}##). So the ##\xi^a## are just the coordinates of ##\mathbb{R}^3##, in this case

    $$ \xi^1 = x, \qquad \xi^2 = y, \qquad \xi^3 = F(x,y). $$
    Note that your formula ought to have partial derivatives:

    $$ g_{ij} = \frac{\partial \xi^a}{\partial x^i} \frac{\partial \xi^b}{\partial x^j} \delta_{ab}. $$
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Find the determinant of the metric on some graph
Loading...