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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 Matrizen Definition $\mathbf {A} ,\ \mathbf {B}$ Summenform: $a_{rc},\ b_{rc}$ Matrixform: ${\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}},\ {\begin{pmatrix}b_{11}&b_{12}\\b_{21}&b_{22}\end{pmatrix}}$ Nullmatrix $\mathbf {0}$ Summenform: $0_{rc}=0$ Matrixform: ${\begin{pmatrix}0&0&0\\0&0&0\\0&0&0\end{pmatrix}}$ Identität $\mathbf {I} =\mathbf {1}$ Summenform: $i_{rc}={\begin{cases}1\ |\ r=c\\0\ |\ r\neq c\end{cases}}$ $i_{jj}=1$ Matrixform: ${\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}$ Addition Addition mit Skalar $\mathbf {C} =s+\mathbf {A} =\mathbf {A} +s$ Summenform: $c_{rc}=s+a_{rc}=a_{rc}+s$ Matrixform: ${\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\\\end{pmatrix}}=s+{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}s+a_{11}&s+a_{12}\\s+a_{21}&s+a_{22}\end{pmatrix}}$ Addition mit Matrix $\mathbf {C} =\mathbf {A} +\mathbf {B} =\mathbf {B} +\mathbf {A}$ Summenform: $c_{rc}=a_{rc}+b_{rc}$ Matrixform: ${\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}={\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}\\a_{21}+b_{21}&a_{22}+b_{22}\end{pmatrix}}$ Multiplikation Multiplikation mit Skalar $\mathbf {C} =s\,\mathbf {A} =\mathbf {A} \,s$ Summenform: $c_{rc}=s\,a_{rc}=a_{rc}\,s$ Matrixform: ${\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}=s\,{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}s\,a_{11}&s\,a_{12}\\s\,a_{21}&s\,a_{22}\end{pmatrix}}$ Komplement $\mathbf {C} =-\mathbf {A} =-1\,\mathbf {A}$ Summenform: $c_{rc}=-1\,a_{rc}=-a_{rc}$ Matrixform: ${\begin{pmatrix}c_{11}&c_{12}\\c_{21}&c_{22}\end{pmatrix}}=-{\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{pmatrix}}={\begin{pmatrix}-a_{11}&-a_{12}\\-a_{21}&-a_{22}\end{pmatrix}}$ Matrixmultiplikation $\mathbf {C} =\mathbf {A} \,\mathbf {B} \neq \mathbf {B} \,\mathbf {A}$ Summenform: $c_{rc}=\sum _{{\color {Red}k}=1}^{m}a_{r{\color {Red}k}}\,b_{{\color {Red}k}c}=a_{r{\color {Red}k}}\,b_{{\color {Red}k}c}$ Matrixform: ${\begin{pmatrix}c_{11}&c_{12}&c_{13}\\c_{21}&c_{22}&c_{23}\\c_{31}&c_{32}&c_{33}\end{pmatrix}}={\begin{pmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\\a_{31}&a_{32}\end{pmatrix}}\,{\begin{pmatrix}b_{11}&b_{12}&b_{13}\\b_{21}&b_{22}&b_{23}\end{pmatrix}}=\dots$ $\dots =\sum _{{\color {Red}k}=1}^{3}{\begin{pmatrix}a_{1{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{1{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{1{\color {Red}k}}\,b_{{\color {Red}k}3}\\a_{2{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{2{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{2{\color {Red}k}}\,b_{{\color {Red}k}3}\\a_{3{\color {Red}k}}\,b_{{\color {Red}k}1}&a_{3{\color {Red}k}}\,b_{{\color {Red}k}2}&a_{3{\color {Red}k}}\,b_{{\color {Red}k}3}\\\end{pmatrix}}=\dots$ $\dots ={\begin{pmatrix}a_{1{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{1{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{1{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{1{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{1{\color {Red}3}}\,b_{{\color {Red}3}3}\\a_{2{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{2{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{2{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{2{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{2{\color {Red}3}}\,b_{{\color {Red}3}3}\\a_{3{\color {Red}1}}\,b_{{\color {Red}1}1}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}1}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}1}&a_{3{\color {Red}1}}\,b_{{\color {Red}1}2}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}2}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}2}&a_{3{\color {Red}1}}\,b_{{\color {Red}1}3}+a_{3{\color {Red}2}}\,b_{{\color {Red}2}3}+a_{3{\color {Red}3}}\,b_{{\color {Red}3}3}\\\end{pmatrix}}$ Multiplikation mit Nullmatrix $\mathbf {C} =\mathbf {0} \,\mathbf {A} =\mathbf {A} \,\mathbf {0} =\mathbf {0}$ Summenform: $c_{rc}=0\,a_{rc}=a_{rc}\,0=0$ Multiplikation mit Identität $\mathbf {C} =\mathbf {1} \,\mathbf {A} =\mathbf {A} \,\mathbf {1} =\mathbf {A}$ Summenform: $c_{rc}=i_{rr}\,a_{rc}=a_{rc}\,i_{rr}=1\,a_{rc}=a_{rc}$ Koummutativität mit skalierter Matrix $\mathbf {A} \,\left(s\,\mathbf {A} \right)=\left(s\,\mathbf {A} \right)\,\mathbf {A}$ Division Inverses Element Definition: $\mathbf {M} ^{-1}\,\mathbf {M} =\mathbf {M} \,\mathbf {M} ^{-1}=1$ Inverse ist eindeutig: $\left(\mathbf {L} \,\mathbf {M} =\mathbf {M} \,\mathbf {L} =1\right)\Rightarrow \mathbf {L} =\mathbf {M} ^{-1}$ Rechenregeln: $\left(\mathbf {M} ^{-1}\right)^{-1}=\mathbf {M}$ $\left(\mathbf {M} \,\mathbf {N} \right)^{-1}=\left(\mathbf {M} ^{-1}\,\mathbf {N} ^{-1}\right)$