Vector fields and are differentiable functions of position in

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

Invariance, Ch Index notation and Cartesian tensors

Problem

Vector fields A(x)\mathbf A(\mathbf x) and B(x)\mathbf B(\mathbf x) are differentiable functions of position in R3\mathbb R^3, referred to orthonormal Cartesian axes. Using the Levi-Civita symbol, its cyclic property ϵijk=ϵkij\epsilon_{ijk}=\epsilon_{kij}, and the contracted identity ϵijkϵimn=δjmδkn−δjnδkm\epsilon_{ijk}\epsilon_{imn}=\delta_{jm}\delta_{kn}-\delta_{jn}\delta_{km}, derive the identity ∇×(A×B)=A(∇⋅B)−B(∇⋅A)+(B⋅∇)A−(A⋅∇)B.\nabla\times(\mathbf A\times\mathbf B) = \mathbf A(\nabla\cdot\mathbf B) - \mathbf B(\nabla\cdot\mathbf A) + (\mathbf B\cdot\nabla)\mathbf A - (\mathbf A\cdot\nabla)\mathbf B.

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