91Ó°ÊÓ

Find ${\mathbf{ds}}^{\mathbf{2}}$in spherical coordinates by the method used to obtain(8.5)for cylindrical coordinates. Use your result to find for spherical coordinates, the scale factors, the vector ds , the volume element, the basis vectors ${\mathbf{a}}_{\mathbf{r}}\mathbf{,}{\mathbf{a}}_{\mathbf{θ}}\mathbf{,}{\mathbf{a}}_{\mathbf{ϕ}}$and the corresponding unit basis vectors${\mathbf{e}}_{\mathbf{r}}\mathbf{,}{\mathbf{e}}_{\mathbf{θ}}\mathbf{,}{\mathbf{e}}_{\mathbf{ϕ}}$ . Write the ${\mathbf{g}}_{\mathbf{ij}}$matrix.

Consider the matrix A in$\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{7}\mathbf{)}\mathbf{}\mathit{o}\mathit{r}\mathbf{}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{10}\mathbf{)}$ .Think of the elements in each row (or column) as the components of a vector. Show that the row vectors form an orthonormal triad (that is each is of unit length and they are all mutually orthogonal), and the column vectors form an orthonormal triad.

Write the transformation equation for a $\mathbf{3}\mathit{r}\mathit{d}$-rank tensor; for a $\mathbf{5}\mathit{t}\mathit{h}$-rank tensor

Parabolic cylinder coordinates $\mathit{u}\mathbf{,}\mathit{v}\mathbf{,}\mathit{z}\mathbf{.}$

$$\begin{gathered} \mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\left(u^2-v^2\right) \\ \mathit{y}\mathbf{=}\mathit{u}\mathit{v} \\ \mathit{z}\mathbf{=}\mathit{z} \end{gathered}$$