Benutzer:MovGP0/Physik/Jacobi-Matrizen

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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 Jacobi-Matrix Koordinatentransformation Transformation Identität $x(r,\theta )=r\cdot \cos \theta$ $x(r(x,y),\,\theta (x,y))\equiv x$ $y(r,\theta )=r\cdot \sin \theta$ $y(r(x,y),\,\theta (x,y))\equiv y$ $r(x,y)={\sqrt {x^{2}+y^{2}}}$ $r(x(r,\theta ),\,y(r,\theta ))\equiv r$ $\theta (x,y)=\arctan {\frac {y}{x}}$ $\theta (x(r,\theta ),\,y(r,\theta ))\equiv \theta$ Erstellung der Jacobi-Matrizen durch Ableitung: ${\begin{pmatrix}x_{r}&x_{\theta }\\y_{r}&y_{\theta }\end{pmatrix}}={\begin{pmatrix}{\frac {\partial }{\partial r}}\,x(r,\theta )&{\frac {\partial }{\partial \theta }}\,x(r,\theta )\\{\frac {\partial }{\partial r}}\,y(r,\theta )&{\frac {\partial }{\partial \theta }}\,y(r,\theta )\end{pmatrix}}=\underbrace {\begin{pmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{pmatrix}} _{\text{Polar}}$ ${\begin{pmatrix}r_{x}&r_{y}\\\theta _{x}&\theta _{y}\end{pmatrix}}={\begin{pmatrix}{\frac {\partial }{\partial x}}\,r(x,y)&{\frac {\partial }{\partial y}}\,r(x,y)\\{\frac {\partial }{\partial x}}\,\theta (x,y)&{\frac {\partial }{\partial y}}\,\theta (x,y)\end{pmatrix}}=\underbrace {\begin{pmatrix}{\frac {x}{\sqrt {x^{2}+y^{2}}}}&{\frac {y}{\sqrt {x^{2}+y^{2}}}}\\{\frac {-y}{x^{2}+y^{2}}}&{\frac {x}{x^{2}+y^{2}}}\end{pmatrix}} _{\text{Kartesisch}}=\underbrace {\begin{pmatrix}{\frac {r\cdot \cos \theta }{r}}&{\frac {r\cdot \sin \theta }{r}}\\{\frac {-r\cdot \sin \theta }{r^{2}}}&{\frac {r\cdot \cos \theta }{r^{2}}}\end{pmatrix}} _{\text{einsetzen}}=\underbrace {\begin{pmatrix}\cos \theta &\sin \theta \\{\frac {-\sin \theta }{r}}&{\frac {\cos \theta }{r}}\end{pmatrix}} _{\text{Polar}}$ Die Transformationen sind invers: ${\begin{pmatrix}x_{r}&x_{\theta }\\y_{r}&y_{\theta }\end{pmatrix}}\cdot {\begin{pmatrix}r_{x}&r_{y}\\\theta _{x}&\theta _{y}\end{pmatrix}}={\begin{pmatrix}x_{r}\cdot r_{x}+x_{\theta }\cdot \theta _{x}&x_{r}\cdot r_{y}+x_{\theta }\cdot \theta _{y}\\y_{r}\cdot r_{x}+y_{\theta }\cdot \theta _{x}&y_{r}\cdot r_{y}+y_{\theta }\cdot \theta _{y}\end{pmatrix}}$ ${\begin{pmatrix}\cos \theta &-r\sin \theta \\\sin \theta &r\cos \theta \end{pmatrix}}\cdot {\begin{pmatrix}\cos \theta &\sin \theta \\{\frac {-\sin \theta }{r}}&{\frac {\cos \theta }{r}}\end{pmatrix}}={\begin{pmatrix}1&0\\0&1\end{pmatrix}}$