Chapter 10: Tensor Analysis

Do Problem 14 for an orthogonal coordinate system with scale factors${\mathit{h}}_{\mathbf{1}}\mathbf{,}{\mathit{h}}_{\mathbf{2}}\mathbf{,}{\mathit{h}}_{\mathbf{3}}$,and compare with the Section 9 formulas

Parabolic cylinder.

Show that if${\mathit{V}}^{\mathbf{i}}$is a contravariant vector then${\mathit{V}}^{\mathbf{i}}\mathbf{=}{\mathit{g}}_{\mathbf{i}\mathbf{j}}{\mathit{V}}^{\mathbf{j}}$is a covariant vector, andthat if${\mathit{V}}^{\mathbf{i}}$is a covariant vector, then${\mathit{V}}^{\mathbf{i}}\mathbf{=}{\mathit{g}}_{\mathbf{i}\mathbf{j}}{\mathit{V}}^{\mathbf{j}}$is a contravariant vector.

${\mathit{X}}_{\mathbf{i}}{\mathit{A}}_{\mathbf{j}}\mathbf{=}{\mathit{B}}_{\mathbf{i}\mathbf{j}}$.

(a) Write the triple scalar productin $\mathbf{A}\cdot (\mathbf{B}\times \mathbf{C})$tensor form and show that it is equal to the determinant in Chapter 6, equation.$\mathbf{(}\mathbf{3}\mathbf{.2}\mathbf{)}$ Hint: See.$\mathbf{(}\mathbf{5}\mathbf{.5}\mathbf{)}$

(b) Write equation$\mathbf{(}\mathbf{3}\mathbf{.2}\mathbf{)}$of Chapter 6 in tensor form to show the equivalence of the various expressions for the triple scalar product. Hint: Change the dummy indices as needed.

In the physics formulas of Problems 9 to 14, identify each symbol as a vector (polar vector) or a pseudovector (axial vector). Use results from the text and the fact that both sides of an equation must have the same tensor character. The definition of the symbols used is:r = displacement, t= time,m= mass, q= electric charge,v = velocity, F= force,$\mathit{\omega }$= angular velocity,$\mathit{\tau }$= torque, L = angular momentum, T = kinetic energy, E= electric field,B= magnetic field. Assume that t, m, andq are scalars. Note that we are working in 3- dimensional physical space and assuming classical (that is nonrelativistic) physics.

$\mathbf{\text{L=mr}}\mathbf{\times }\mathit{v}\mathbf{=}\mathit{m}\mathit{r}\mathbf{\times }\mathbf{(}\mathit{\omega }\mathbf{\times }\mathit{r}\mathbf{)}$

Show that the nine quantities ${\mathbf{T}}_{\mathbf{ij}}\mathbf{=}\mathbf{(}\mathbf{\partial }{\mathbf{V}}_{\mathbf{i}}\mathbf{)}\mathbf{/}\mathbf{(}\mathbf{\partial }{\mathbf{x}}_{\mathbf{J}}\mathbf{)}$ (which are the Cartesian components of $\mathbf{\nabla }\mathbf{V}$ where V is a vector) satisfy the transformation equations $\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{14}\mathbf{)}\mathbf{}$for a Cartesian ${\mathbf{2}}^{\mathbf{nd}}\mathbf{}$ -rank tensor. Show that they do not satisfy the general tensor transformation equations as in $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{12}\mathbf{)}$ . Hint: Differentiate $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{9}\mathbf{)}$or$\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{10}\mathbf{)}\mathbf{}$partially with respect to, say,$\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{k}\mathbf{}}$. You should get the expected terms [as in$\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{12}\mathbf{)}\mathbf{}$ ] plus some extra terms; these extraneous terms show that$\mathbf{(}\mathbf{\partial }{\mathbf{V}}_{\mathbf{i}}\mathbf{)}\mathbf{/}\mathbf{(}\mathbf{\partial }{\mathbf{x}}_{\mathbf{J}}\mathbf{)}\mathbf{}$ is not a tensor under general transformations. Comment: It is possible to express the components of$\mathbf{\nabla }\mathbf{V}$ correctly in general coordinate systems by taking into account the variation of the basis vectors in length and direction.

In (10.18), show by raising and lowering indices that ${\mathit{a}}_{\mathbf{i}}{\mathit{V}}^{\mathbf{i}}\mathbf{=}{\mathit{a}}^{\mathbf{i}}{\mathit{V}}_{\mathbf{i}}$ . Also, write (10.18) for an orthogonal coordinate system with ${\mathit{g}}_{\mathbf{i}\mathbf{j}}$and${\mathit{g}}^{\mathbf{i}\mathbf{j}}$written in terms of the scale factors.

:Do Problem 5 for the coordinate systems indicated in Problems 10 to 13.Eliptical cylinder.

${\mathit{X}}_{\mathbf{i}\mathbf{j}}{\mathit{A}}_{\mathbf{k}}\mathbf{=}{\mathit{B}}_{\mathbf{i}\mathbf{j}\mathbf{k}}$.