I have the following problem:
Compute the integral $\int_M f(x)dS(x)$ where $f(x,y)=xy$ and M is the boundary of the triangle with vertices $(0,0),(1,2),(2,1)$.
In the lecture we used the definition of an integral over a manifold to solve such a problem. Therefore I spited up my set M in $$M_1=\{\frac{x}{2}|0\leq x\leq 2\}, M_2=\{-x+3|1\leq x\leq 2\}, M_3=\{2x|0\leq x\leq 1\}$$ Then M is clearly the disjoint union of all this $M_i$'s. Now I defined my 3 charts as follows
- $M_1$: $\phi_1(x)=(x,\frac{x}{2}),0\leq x\leq 2 $
- $M_2$: $\phi_2(x)=(x,-x+3), 1\leq x\leq 2$
- $M_3$: $\phi_3(x)=(x,2x), 0\leq x\leq 1$
Now I used that $$\int_M f(x)dS(x)=\sum_{i=1}^3 \int_{M_i}f(\phi_i(t)\cdot \sqrt{g_{\phi_i} (t)}dt$$ where $g_{\phi_i}(t)$ is the determinant of the gram matrix. But then I somehow get $$g_{\phi_1}=0.75. g_{\phi_2}=0, g_{\phi_3}=-3$$ But this somehow doesn't work.
Could please someone help me finding my error?
It would be nice if we can do it in my way, since this is the only way we had in the lecture and thus the only one we can use at the exam.
Thank you a lot.
If $M\subset \mathbb R^n$ is a $k$-dimensional submanifold and $\phi:\Omega\to M$, $\Omega\subset\mathbb R^k$ a parametrization then the integral of some function $f:M\to\mathbb R$ is defined by $\int_M f=\int_\Omega (f\circ\phi)\cdot \sqrt{\det(D\phi^T D\phi)}$.
In general $D\phi^T D\phi$ is a $k\times k$ matrix. If $k=1$, then $D\phi^T D\phi=[\phi'^T\phi']=[|\phi'|^2]$, which is a $1\times1$ matrix and $\sqrt{\det(D\phi^T D\phi)}=|\phi'|$. This yields the usual formula for the line integral along some curve $\gamma:[a,b]\to\mathbb R^n$ : $\int_\gamma f=\int_a^bf(\gamma(t))\cdot|\gamma'(t)|dt.$
For example the first part of your integral is $\int_{M_1}f(\phi_1(t)\cdot \sqrt{g_{\phi_1} (t)}dt=\int_0^2f(t,\frac t 2)\cdot|(1,\frac 12)|dt=\int_0^2\frac 12 t^2\cdot\frac {\sqrt {5}} {2}dt$.
Hope this helps.