I have problems solving this integral since 2 days -.-
$\int_{S^2} f \cdot n \; dS \; \; \;$ with
$f(x,y,z):=(y^3, z^3, x^3)^T$, $\; \; S^2=\{(x,y,z) \in \mathbb{R^3} | x^2+y^2+z^2 = 1\}$ and $n$ ist the outer normal vector.
We -theoretically- "learned" the theorem of Stokes and may use it.
Thanks for any help :-)
Edit1: Stokes was: $\int_{\partial S} f = \int_{S}$d$f$? The professor never uses "$n\cdot f$" but the guy publishing the tasks does... confusing like hell. Thanks for unraveling this one! What does your $\land$ mean?
You don't want Stokes' Theorem. You want the Divergence Theorem, which is usually stated in Calculus classes for $\mathbb{R}^{3}$ vector functions $\vec{f}$ as follows: Let $S$ be a smooth closed surface enclosing a volume $V$. Let $\vec{f}$ be a continuously differentiable vector function on an open set containing $S$ and $V$ (this is simplest to state.) Then $$ \int_{S} \vec{f}\cdot d\vec{S} = \int_{V}\nabla\cdot \vec{f}\,dV. $$ On the left, $\int_{S}\vec{f}\cdot d\vec{S}$ means $\int_{S}\vec{f}\cdot \hat{n}\,dS$; $\vec{n}$ is the unit outward normal to $S$; and $dS$ means the area measure on $S$. On the right side, $dV$ means the usual volume measure in $V$, and $\nabla\cdot$ is the divergence operator.