1.2 – Ausführliche Darstellung von Ausdrücken in einsteinscher Summenkonvention

 

Schreiben Sie folgende Ausdrücke in möglichst ausführlicher Form:

  1. \frac{{\partial {\tau _{ij}}}}{{\partial {x_i}}}
  2. \frac{{\partial \left( {{\tau _{ij}}{v_j}} \right)}}{{\partial {x_i}}}

Lösung

a)

\frac{{\partial {\tau _{ij}}}}{{\partial {x_i}}} = \sum\limits_{i = 1}^n {\frac{{\partial {\tau _{ij}}}}{{\partial {x_i}}}} = \frac{{\partial {\tau _{1j}}}}{{\partial {x_1}}}+\frac{{\partial {\tau _{2j}}}}{{\partial {x_2}}}+ \ldots +\frac{{\partial {\tau _{nj}}}}{{\partial {x_n}}}

b)

\frac{{\partial \left( {{\tau _{ij}}{v_j}} \right)}}{{\partial {x_i}}} = \sum\limits_{i = 1}^m {\frac{{\partial \left( {\sum\limits_{j = 1}^n {{\tau _{ij}}{v_j}} } \right)}}{{\partial {x_i}}}} = \frac{{\partial \left( {{\tau _{11}}{v_1}+ \ldots +{\tau _{1n}}{v_n}} \right)}}{{\partial {x_1}}}+ \ldots +\frac{{\partial \left( {{\tau _{m1}}{v_m}+ \ldots +{\tau _{mn}}{v_n}} \right)}}{{\partial {x_m}}}