Benutzer:MovGP0/Physik/Tensornotation

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Physik Mathematische Grundlagen Mathematik für ET Semester 1 • Semester 2 • Semester 3 Mathematik Kombinatorik und Permutationen • Ableitung • Dreiecksnotation • Gradient, Divergenz, Rotation • Metrischer Tensor • Christoffelsymbole • Krümmungstensor • Energie-Impuls-Tensor • Kosmologische Konstante Quantenmechanik mit Matrizen Imaginäre und komplexe Zahlen • Vektoren • Matrizen • Jacobi-Matrizen • Pauli-Matrizen • Hermitesche Matrizen Lineare Algebra und Tensoren Tensornotation • Tensoren • Dirac-Notation • Körper • Vektorraum Gruppentheorie Gruppen, Lie Gruppen, Lie Algebren Tensornotation Seien die unabhängigen Variablen $a^{1},a^{2},a^{3},\ldots ,a^{n}=a^{i}$ gegeben durch: ${\begin{matrix}a^{1}=a^{1}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m})\\a^{2}=a^{2}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m})\\a^{3}=a^{3}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m})\\\vdots \\a^{n}=a^{n}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m})\end{matrix}}$ damit gilt $f=F(a^{1},a^{2},a^{3},\ldots ,a^{n})=F(a^{i})$ bzw. $f(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m})=F(a^{1}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m}),a^{2}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m}),a^{3}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m}),\ldots ,a^{n}(\mu ^{1},\mu ^{2},\mu ^{3},\ldots ,\mu ^{m}))=F(a^{i}(\mu ^{j}))$ Ableitung: ${\begin{aligned}{\frac {\partial f}{\partial \mu ^{j}}}=&{\frac {\partial F}{\partial a^{1}}}{\frac {\partial a^{1}}{\partial \mu ^{1}}}+{\frac {\partial F}{\partial a^{2}}}{\frac {\partial a^{2}}{\partial \mu ^{1}}}+{\frac {\partial F}{\partial a^{3}}}{\frac {\partial a^{3}}{\partial \mu ^{1}}}+\ldots +{\frac {\partial F}{\partial a^{n}}}{\frac {\partial a^{n}}{\partial \mu ^{1}}}\\&+{\frac {\partial F}{\partial a^{1}}}{\frac {\partial a^{1}}{\partial \mu ^{2}}}+{\frac {\partial F}{\partial a^{2}}}{\frac {\partial a^{2}}{\partial \mu ^{2}}}+{\frac {\partial F}{\partial a^{3}}}{\frac {\partial a^{3}}{\partial \mu ^{2}}}+\ldots +{\frac {\partial F}{\partial a^{n}}}{\frac {\partial a^{n}}{\partial \mu ^{2}}}\\&+{\frac {\partial F}{\partial a^{1}}}{\frac {\partial a^{1}}{\partial \mu ^{3}}}+{\frac {\partial F}{\partial a^{2}}}{\frac {\partial a^{2}}{\partial \mu ^{3}}}+{\frac {\partial F}{\partial a^{3}}}{\frac {\partial a^{3}}{\partial \mu ^{3}}}+\ldots +{\frac {\partial F}{\partial a^{n}}}{\frac {\partial a^{n}}{\partial \mu ^{3}}}\\&+\ldots \\&+{\frac {\partial F}{\partial a^{1}}}{\frac {\partial a^{1}}{\partial \mu ^{m}}}+{\frac {\partial F}{\partial a^{2}}}{\frac {\partial a^{2}}{\partial \mu ^{m}}}+{\frac {\partial F}{\partial a^{3}}}{\frac {\partial a^{3}}{\partial \mu ^{m}}}+\ldots +{\frac {\partial F}{\partial a^{n}}}{\frac {\partial a^{n}}{\partial \mu ^{m}}}&=\sum _{i}{{\frac {\partial F}{\partial a^{i}}}{\frac {\partial a^{i}}{\partial \mu ^{j}}}}=\underbrace {{\frac {\partial F}{\partial a^{i}}}{\frac {\partial a^{i}}{\partial \mu ^{j}}}} _{\text{Einstein-Notation}}\end{aligned}}$