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| -rw-r--r-- | content/analysis_3.tex | 12 |
1 files changed, 12 insertions, 0 deletions
diff --git a/content/analysis_3.tex b/content/analysis_3.tex index 798b7c9..ad812cf 100644 --- a/content/analysis_3.tex +++ b/content/analysis_3.tex @@ -508,6 +508,18 @@ die \emph{Gramsche Determinante} von $F$. \vspace{2mm} +Für $m = 3$ gilt insbesondere: + +\vspace{-4mm} +\begin{align*} +g_F(t) &= |\partial_1F(t) \times \partial_2F(t)|_2^2\\ + &= \left|\begin{pmatrix} + \partial_1 F_2(t) \partial_2 F_3(t) - \partial_1 F_3(t) \partial_2 F_2(t) \\ + \partial_1 F_3(t) \partial_2 F_1(t) - \partial_1 F_1(t) \partial_2 F_3(t) \\ + \partial_1 F_1(t) \partial_2 F_2(t) - \partial_1 F_2(t) \partial_2 F_1(t) +\end{pmatrix}\right|_2^2 +\end{align*} + Im Graphenfall $F(t) = (t,h(t))$ für $t \in U$, $U \subseteq \R^{m-1}$ offen und $h \in C^1(U,\R)$ gilt: $\sqrt{g_F(t)} = \sqrt{1+|\nabla h(t)|_2^2}$ |