I've got a couple of doubts in the following excercises. Sorry if somethings wrong or hard to understand in my writing, I'm non-native speaker.
- Calculate de integral of the function:
$$f(x,y) = \sqrt{x^2+y^2}\exp(\sqrt{x^2+y^2}-r)$$
over the set $A=({(x,y)\in R^2| 0 \leqslant x^2+y^2\leqslant r})$ for $r>0$
I've performed a variable change to cilindrical coordinates so
$\int \sqrt{x^2+y^2}e^{\sqrt{x^2+y^2}-r}\, dx\,dy = \int \sqrt{r^2} e^{\sqrt{r^2}-r'} r\, dr\, d\theta$ considering $r \ne r'$ Am I I getting this done by the right way, is there anything that I'm not considering?
- Let $\vec{F}:\Omega\subset R^3 \rightarrow R^3$ class $C^{1}$. Prove that the surface integral of $\nabla \times \vec{F}$ over $\Sigma$ its equal $0$, where $\Sigma \subset \Omega$ it's a sphere (which centre not necessary in $\Omega$) and parameterization of $\Sigma$ it's normal outside,
I've got no clue on how to perform this proof, I've tested performing over Stokes Theorem but not really getting somewhere. Any ideas or even the proof? It's the bonus of my last evaluation on my course of multivariable integral calculus.
Since $r$ is doing double duty lets lets use $\rho$ as the radius in the coordinate system and $R$ as the radius of the cylinder.
$x = \rho \cos \theta\\ y = \rho \sin \theta\\ 0\le\rho \le R$
$\int_0^{2\pi}\int_0^{R} \rho^2e^{\rho-R}\ d\rho\ d\theta\\ 2\pi(e^{\rho-R})(\rho^2-2\rho + 2)|_0^R\\ 2\pi(R^2 - 2R + 2 - 2e^{-R})$
8) By the divergence theorem $\iint G\ dS = \iiint \nabla\cdot G \ dS\\ G = \nabla \times F\\ \nabla \cdot G = \nabla \cdot (\nabla \times F) = 0$