Semester 1 • Semester 2 • Semester 3
Kombinatorik und Permutationen • Ableitung • Dreiecksnotation • Gradient, Divergenz, Rotation • Metrischer Tensor • Christoffelsymbole • Krümmungstensor • Energie-Impuls-Tensor • Kosmologische Konstante
Imaginäre und komplexe Zahlen • Vektoren • Matrizen • Jacobi-Matrizen • Pauli-Matrizen • Hermitesche Matrizen
Tensornotation • Tensoren • Dirac-Notation • Körper • Vektorraum
Gruppen, Lie Gruppen, Lie Algebren
Erstellung der Jacobi-Matrizen durch Ableitung:
( x r x θ y r y θ ) = ( ∂ ∂ r x ( r , θ ) ∂ ∂ θ x ( r , θ ) ∂ ∂ r y ( r , θ ) ∂ ∂ θ y ( r , θ ) ) = ( cos θ − r sin θ sin θ r cos θ ) ⏟ Polar {\displaystyle {\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}}}
( r x r y θ x θ y ) = ( ∂ ∂ x r ( x , y ) ∂ ∂ y r ( x , y ) ∂ ∂ x θ ( x , y ) ∂ ∂ y θ ( x , y ) ) = ( x x 2 + y 2 y x 2 + y 2 − y x 2 + y 2 x x 2 + y 2 ) ⏟ Kartesisch = ( r ⋅ cos θ r r ⋅ sin θ r − r ⋅ sin θ r 2 r ⋅ cos θ r 2 ) ⏟ einsetzen = ( cos θ sin θ − sin θ r cos θ r ) ⏟ Polar {\displaystyle {\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:
( x r x θ y r y θ ) ⋅ ( r x r y θ x θ y ) = ( x r ⋅ r x + x θ ⋅ θ x x r ⋅ r y + x θ ⋅ θ y y r ⋅ r x + y θ ⋅ θ x y r ⋅ r y + y θ ⋅ θ y ) {\displaystyle {\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}}}
( cos θ − r sin θ sin θ r cos θ ) ⋅ ( cos θ sin θ − sin θ r cos θ r ) = ( 1 0 0 1 ) {\displaystyle {\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}}}
