studying the finite element method, I had to calculate a line integral for a two variable function, or a flux for three-variable function.
After a little bit of reaserch, I found these formulas :
$$\int_{\Gamma} f(x,y)ds \; \stackrel{def}{\equiv}\; \int_{0}^1 f(\vec\gamma(t)) \; ||\vec \gamma'(t)|| \; dt $$ and $$\iint_{d\Omega} f(x,y,z)dS \; \stackrel{def}{\equiv} \; \iint_D f(\vec\phi(u,v)) \; || \frac{\partial \vec \phi}{\partial u} \times \frac{\partial \vec \phi}{\partial v} || \;dudv $$
where $\vec\gamma$ and $\vec\phi$ are a parametrisation of $\Gamma$ and $d\Omega$.
Now, I have to perform a change of variables, for example in the first integral. The problem is that I don't know how to do that.
I would like to write something like : $$ " \int_{\psi(\Gamma)} f(x,y)ds \; = \int_{\Gamma} (f \circ \psi) (x',y') \; |Jac \;\psi| \; ds' \; "$$
because it would look like a classic change of variables in $\mathbb{R^2}$, but I am not sure that this is right.
Can someone tell me the right formula for such a change of variables ?
Regards,
Mario