Paper 3, Section I, B

Vector Calculus | Part IA, 2021

(a) What is meant by an antisymmetric tensor of second rank? Show that if a second rank tensor is antisymmetric in one Cartesian coordinate system, it is antisymmetric in every Cartesian coordinate system.

(b) Consider the vector field F=(y,z,x)\mathbf{F}=(y, z, x) and the second rank tensor defined by Tij=Fi/xjT_{i j}=\partial F_{i} / \partial x_{j}. Calculate the components of the antisymmetric part of TijT_{i j} and verify that it equals (1/2)ϵijkBk-(1 / 2) \epsilon_{i j k} B_{k}, where ϵijk\epsilon_{i j k} is the alternating tensor and B=×F\mathbf{B}=\boldsymbol{\nabla} \times \mathbf{F}.

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