1.1 – Einsteinsche Summenkonvention, Gradient und Rotation

 

Man zeige:

{u_i}\frac{{\partial {u_j}}}{{\partial {x_i}}} = \left( {\vec u \cdot \operatorname{grad} } \right)\vec u = \vec \omega \times \vec u+\operatorname{grad} \left( {\frac{1}{2}{{\vec u}^2}} \right)

Lösung

Einsteinsche Summenkonvention:

{a_i}{x_i}: = \sum\limits_{i = 1}^3 {{a_i}{x_i}} = {a_1}{x_1}+{a_2}{x_2}+{a_3}{x_3}

{A_{ij}}{x_j} = \sum\limits_{j = 1}^3 {{A_{ij}}{x_j}} = {A_{i1}}{x_i}+{A_{i2}}{x_2}+{A_{i3}}{x_3}

Tritt in einem Ausdruck ein Index zwei Mal auf, so wird der Ausdruck über alle vorgesehenen Werte dieses Index summiert. Tritt ein Index in den Ausdrücken einer Gleichung nur einmal auf, so bedeutet das, dass die betreffende Gleichung für alle Werte gilt, die der Index durchlaufen kann.

\vec \omega ist definiert als die Rotation \vec \omega : = \nabla \times \vec u.

Einsetzen:

\left( {\nabla \times \vec u} \right) \times \vec u+\operatorname{grad} \left( {\frac{1}{2}{{\vec u}^2}} \right)

= \left( {\begin{array}{*{20}{c}}{\frac{{\partial {u_3}}}{{\partial y}}-\frac{{\partial {u_2}}}{{\partial z}}} \\{\frac{{\partial {u_1}}}{{\partial z}}-\frac{{\partial {u_3}}}{{\partial x}}} \\{\frac{{\partial {u_2}}}{{\partial x}}-\frac{{\partial {u_1}}}{{\partial y}}} \\ \end{array} } \right) \times \left( {\begin{array}{*{20}{c}}{{u_1}} \\{{u_2}} \\{{u_3}} \\ \end{array} } \right)+\operatorname{grad} \left( {\frac{1}{2}\left( {{u_1}{u_1}+{u_2}{u_2}+{u_3}{u_3}} \right)} \right)

= \left( {\begin{array}{*{20}{c}}{\left( {\frac{{\partial {u_1}}}{{\partial z}}-\frac{{\partial {u_3}}}{{\partial x}}} \right){u_3}-\left( {\frac{{\partial {u_2}}}{{\partial x}}-\frac{{\partial {u_1}}}{{\partial y}}} \right){u_2}} \\{\left( {\frac{{\partial {u_2}}}{{\partial x}}-\frac{{\partial {u_1}}}{{\partial y}}} \right){u_1}-\left( {\frac{{\partial {u_3}}}{{\partial y}}-\frac{{\partial {u_2}}}{{\partial z}}} \right){u_3}} \\{\left( {\frac{{\partial {u_3}}}{{\partial y}}-\frac{{\partial {u_2}}}{{\partial z}}} \right){u_2}-\left( {\frac{{\partial {u_1}}}{{\partial z}}-\frac{{\partial {u_3}}}{{\partial x}}} \right){u_1}} \\ \end{array} } \right)+\left( {\begin{array}{*{20}{c}}{{u_1}\frac{{\partial {u_1}}}{{\partial x}}+{u_2}\frac{{\partial {u_2}}}{{\partial x}}+{u_3}\frac{{\partial {u_3}}}{{\partial x}}} \\{{u_1}\frac{{\partial {u_1}}}{{\partial y}}+{u_2}\frac{{\partial {u_2}}}{{\partial y}}+{u_3}\frac{{\partial {u_3}}}{{\partial y}}} \\{{u_1}\frac{{\partial {u_1}}}{{\partial z}}+{u_2}\frac{{\partial {u_2}}}{{\partial z}}+{u_3}\frac{{\partial {u_3}}}{{\partial z}}} \\ \end{array} } \right)

= \left( {\begin{array}{*{20}{c}}{{u_3}\frac{{\partial {u_1}}}{{\partial z}}-{u_3}\frac{{\partial {u_3}}}{{\partial x}}-{u_2}\frac{{\partial {u_2}}}{{\partial x}}+{u_2}\frac{{\partial {u_1}}}{{\partial y}}+{u_1}\frac{{\partial {u_1}}}{{\partial x}}+{u_2}\frac{{\partial {u_2}}}{{\partial x}}+{u_3}\frac{{\partial {u_3}}}{{\partial x}}} \\{{u_1}\frac{{\partial {u_2}}}{{\partial x}}-{u_1}\frac{{\partial {u_1}}}{{\partial y}}-{u_3}\frac{{\partial {u_3}}}{{\partial y}}+{u_3}\frac{{\partial {u_2}}}{{\partial z}}+{u_1}\frac{{\partial {u_1}}}{{\partial y}}+{u_2}\frac{{\partial {u_2}}}{{\partial y}}+{u_3}\frac{{\partial {u_3}}}{{\partial y}}} \\{{u_2}\frac{{\partial {u_3}}}{{\partial y}}-{u_2}\frac{{\partial {u_2}}}{{\partial z}}-{u_1}\frac{{\partial {u_1}}}{{\partial z}}+{u_1}\frac{{\partial {u_3}}}{{\partial x}}+{u_1}\frac{{\partial {u_1}}}{{\partial z}}+{u_2}\frac{{\partial {u_2}}}{{\partial z}}+{u_3}\frac{{\partial {u_3}}}{{\partial z}}} \\ \end{array} } \right)

= \left( {\begin{array}{*{20}{c}}{{u_1}\frac{{\partial {u_1}}}{{\partial x}}+{u_2}\frac{{\partial {u_1}}}{{\partial y}}+{u_3}\frac{{\partial {u_1}}}{{\partial z}}} \\{{u_1}\frac{{\partial {u_2}}}{{\partial x}}+{u_2}\frac{{\partial {u_2}}}{{\partial y}}+{u_3}\frac{{\partial {u_2}}}{{\partial z}}} \\{{u_1}\frac{{\partial {u_3}}}{{\partial x}}+{u_2}\frac{{\partial {u_3}}}{{\partial y}}+{u_3}\frac{{\partial {u_3}}}{{\partial z}}} \\ \end{array} } \right)

= {u_i}\frac{{\partial {u_j}}}{{\partial {x_i}}}