Vectors have Cartesian components

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

Invariance, Ch Index notation and Cartesian tensors

Problem

Vectors A,B,C,D\mathbf A,\mathbf B,\mathbf C,\mathbf D have Cartesian components Ai,Bi,Ci,DiA_i,B_i,C_i,D_i. Using the Levi-Civita symbol and the contracted identity ϵijkϵimn=δjmδkn−δjnδkm\epsilon_{ijk}\epsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}, derive Lagrange's identity (A×B)⋅(C×D)=(A⋅C)(B⋅D)−(A⋅D)(B⋅C).(\mathbf A\times\mathbf B)\cdot(\mathbf C\times\mathbf D) = (\mathbf A\cdot\mathbf C)(\mathbf B\cdot\mathbf D) - (\mathbf A\cdot\mathbf D)(\mathbf B\cdot\mathbf C).

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