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

Apply Gauss's theorem when the metric is unknown

  1. Oct 18, 2012 #1
    Let $$f:U \to \mathbb{R}^3$$ be a surface, where $$U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$$ Consider the two closed square regions $$4F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, |u^2|\leq1\}$$ and $$F_2=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq2, |u^2|\leq2\}.$$

    While the first fundamental form $$g$$ is unknown in the inner region $$F_1$$, it is the Euclidean metric outside of $$F_1$$:$$g=(\mathrm{d}u^1)^2+(\mathrm{d}u^2)^2$$ at all $$u\in U-F_1$$
    Also, it is known that the vector field $$X=u^1\frac{\partial}{\partial u^1}+u^2\frac{\partial}{\partial u^2}$$ satiesfies $$\mathrm{div}X=2$$ for ALL points of U.
    Show that the total surface area of $$f(F_2)$$ is 16.

    Remark: Recall that the Gauss theorem states: Let $$M$$ be an oriented surface with a Riemannian metric. Let $$X$$ be a vector field on $$M$$. Then for each polygon on $$M$$ given by $$P:F \to M$$ it has $$\int_{P(F)}(\mathrm{div}X)\mathrm{d}M=\int_{\partial P(F)}l_{X}\mathrm{d}M$$,where $$l_{X}\mathrm{d}M=-X^2\sqrt|g|\mathrm{d}u^1+X^1\sqrt|g|\mathrm{d}u^2$$.

    ,Setting $$f_1:=-X^2\sqrt|g|$$ and $$f_2:=X^1\sqrt|g|$$ to get $$\int_{P(F)}(\mathrm{div}X)\mathrm{d}M=\iint_F}(\frac{\partial f_2}{\partial u^1}-\frac{\partial f_1}{\partial u^2})\mathrm{d}u^1\mathrm{d}u^2.$$

    In this problem $|g|=1$ outside $F_1$ but it's unknown on $F_1$.

    We need to compute $$\int_{f(F_2)}\mathrm{d}M=\frac{1}{2}\int_{f(F_2)}(\mathrm{div}X)\mathrm{d}M$$. By the convention of notations it has $X^i=u^i$ and thus $$\frac{1}{2}\int_{f(F_2)}(\mathrm{div}X)\mathrm{d}M=\frac{1}{2}\int_{-2}^{2}\int_{-2}^{2}(1-(-1))\mathrm{d}u^1\mathrm{d}u^2=16.$$

    BUT my computation assumes $$g$$ is also the Euclidean metric on $$F_1$$, which may not be the case. How to modify my computation for it to be fully correct?
    Last edited: Oct 18, 2012
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted

Similar Discussions: Apply Gauss's theorem when the metric is unknown
  1. Metric/metric tensor? (Replies: 1)

  2. Gauss Curvature (Replies: 1)

  3. Gauss Bonnet Theorem (Replies: 1)