Using the summation convention

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Tensors and differential geometry · Tensors and differential geometry

Action and Symmetry, Ch Non-inertial frames from one Lagrangian

Problem

Using the summation convention,

\delta_{ij}

and

\epsilon_{ijk}

, prove that for any three vectors

\mathbf a

\mathbf b

\mathbf c

in three dimensions,

\mathbf a\cdot(\mathbf b\times\mathbf c) = \mathbf c\cdot(\mathbf a\times\mathbf b),

by writing both sides in index notation and using the cyclic symmetry of the Levi-Civita symbol,

\epsilon_{ijk}=\epsilon_{kij}=\epsilon_{jki}

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