Mathematik-Online-Lexikon: Beispiel: Flussintegral

$\begin{displaymath} S:\quad \vec{r}(u,v)=\left( \begin{array}{c} u^2 \\ u+v\\ v^2\\ \end{array}\right),\quad 0\leq u,v \leq 1 \end{displaymath}$

$\begin{displaymath} \partial_u \vec{r}(u,v)=\left( \begin{array}{c} 2u\\ 1\\ 0\\... ...(u,v)=\left( \begin{array}{c} 0\\ 1\\ 2v\\ \end{array}\right) \end{displaymath}$

$\begin{displaymath} \vec{n}(u,v) = \partial_u \vec{r}(u,v)\times \partial_v \ve... ...left( \begin{array}{c} 2v\\ -4uv\\ 2u\\ \end{array}\right)\,. \end{displaymath}$

$\displaystyle \iint\limits_{S} \vec{F} \cdot d\vec{S}$	$\displaystyle =\int\limits_0^1\int\limits_0^1 \left( \begin{array}{c} u^2\\ 1\\... ...\right)\cdot\left( \begin{array}{c} 2v\\ -4uv\\ 2u\\ \end{array}\right)\,du\,dv$
	$\displaystyle =\int\limits_0^1\int\limits_0^1 2u^2v-4uv+2u^2v^2+2uv^3 \,du\,dv$
	$\displaystyle = \int\limits_0^1 \left[ \frac{2}{3}u^3v-2u^2v+\frac{2}{3}u^3v^2+u^2v^3 \right]_0^1\, dv$
	$\displaystyle = \int\limits_0^1-\frac{4}{3}v+\frac{2}{3}v^2+v^3\,dv = \left[-\frac{2}{3}v^2+\frac{2}{9}v^3+\frac{1}{4}v^4\right]_0^1$
	$\displaystyle =-\frac{7}{36}\,.$

	[Home] [Lexikon] [Aufgaben] [Tests] [Kurse] [Begleitmaterial] [Hinweise] [Mitwirkende] [Publikationen]
	Mathematik-Online-Lexikon:
	Beispiel: Flussintegral