4B (Vector Calculus)

A second-rank tensor is antisymmetric when

T_{ij} = - T_{ji} .

Under an orthogonal Cartesian coordinate change with matrix

R

T_{ij}^{'} = R_{ik} R_{j ℓ} T_{k ℓ} .

Therefore

T_{ji}^{'} = R_{jk} R_{i ℓ} T_{k ℓ} = - R_{jk} R_{i ℓ} T_{ℓ k} = - T_{ij}^{'},

after relabelling the dummy indices. Hence the antisymmetric second-rank tensor property is independent of Cartesian coordinates.

b

For

F = (y, z, x)

T_{ij} = \frac{\partial F _{i}}{\partial x _{j}} = 001100010 .

Its antisymmetric part is

\frac{1}{2} (T - T^{T}) = 0 - 1/2 1/2 1/2 0 - 1/2 - 1/2 1/2 0 .

Also,

B = \nabla \times F = (- 1, - 1, - 1) .

For example,

- (1/2) ϵ_{12 k} B_{k} = 1/2

, and the same componentwise calculation gives

\frac{1}{2} (T_{ij} - T_{ji}) = - \frac{1}{2} ϵ_{ijk} B_{k} .

There are no discussions about this article yet.

There are currently no matching articles.