Paper 3, Section I, C

Vector Calculus | Part IA, 2016

If v=(v1,v2,v3)\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right) and w=(w1,w2,w3)\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right) are vectors in R3\mathbb{R}^{3}, show that Tij=viwjT_{i j}=v_{i} w_{j} defines a rank 2 tensor. For which choices of the vectors v\mathbf{v} and w\mathbf{w} is TijT_{i j} isotropic?

Write down the most general isotropic tensor of rank 2 .

Prove that ϵijk\epsilon_{i j k} defines an isotropic rank 3 tensor.

Typos? Please submit corrections to this page on GitHub.