91Ó°ÊÓ

If \(A\) and \(B\) are two \(N \times N\) matrices, show that \((A B)^{T}=B^{T} A^{T}\) where \(A^{T}\) is the transpose of \(A\) defined by interchanging the rows and columns of \(A\).

Write in suffix notation the vector equation \(a \times b+c=(a \cdot b) b-d\).

Simplify the suffix notation expressions (a) \(\delta_{i j} \delta_{j k} \delta_{k i}\) (b) \(\epsilon_{i j k} \epsilon_{k l m} \epsilon_{m n i} .\)

Simplify the suffix notation expression \(\delta_{i j} a_{j} b_{l} c_{k} \delta_{l i}\) and write the result in vector form.

The vector fields \(\boldsymbol{u}\) and \(\boldsymbol{w}\) and the scalar field \(\phi\) are related by the equation $$ \boldsymbol{u}+\boldsymbol{\nabla} \times \boldsymbol{w}=\boldsymbol{\nabla} \phi+\nabla^{2} \boldsymbol{u} $$ and \(u\) is solenoidal. Show that \(\phi\) obeys Laplace's equation.