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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
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