Chapter 10: Tensor Analysis

In the text and problems so far, we have found the e vectors for Question: Using the results of Problem 1, express the vector in Problem 4 in spherical coordinates.

Write the tensor transformation equations for ${\mathit{\epsilon }}_{\mathbf{i}\mathbf{j}\mathbf{k}}{\mathit{\epsilon }}_{\mathbf{m}\mathbf{n}\mathbf{p}}$to show that this is a (rank 6) tensor (nota pseudo tensor). Hint:Write (6.1) for each$\mathit{\epsilon }$and multiply them, being careful not to re-use a pair of summation indices.

Write out $\mathbf{\nabla }\mathit{U}\mathbf{,}\mathbf{\nabla }\mathbf{.}\mathit{V}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathbf{.}\mathit{U}\mathbf{,}\mathbf{\nabla }\mathbf{\times }\mathit{V}$in spherical coordinates

Write equations (2.12) out in detail and solve the three simultaneous equations (say by determinants) for${\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}$in terms of${\mathbf{x}}_{\mathbf{1}}\mathbf{\text{'}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{\text{'}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}\mathbf{\text{'}}$to verify equations (2.13) . Use your results in Problem 4.

Show by the quotient rule (Section 3 ) that ${\mathbf{C}}_{\mathbf{ijkm}}$in $\mathbf{(}\mathbf{7}\mathbf{.}\mathbf{5}\mathbf{)}$is a ${\mathbf{4}}^{\mathbf{th}}$-rank tensor.

Let${\mathit{T}}_{\mathbf{j}\mathbf{k}\mathbf{m}\mathbf{n}}$ be the tensor in $\mathbf{(}\mathbf{5}\mathbf{.8}\mathbf{)}$. This is a${\mathbf{4}}^{\mathbf{t}\mathbf{h}}$ -rank tensor and so has ${\mathbf{3}}^{\mathbf{4}}\mathbf{=}\mathbf{81}$ components. Most of the components are zero. Find the nonzero components and their values. Hint: See discussion after$\mathbf{(}\mathbf{5}\mathbf{.8}\mathbf{)}$ .

Write$\mathbf{\nabla }\mathit{u}$in polar coordinates in terms of its physical components and the unitbasis vectors${\mathit{e}}_{\mathbf{i}}$, and in terms of its covariant components and the contravariantbasis vectors${\mathit{a}}_{\mathbf{i}}$. What is the relation between the contravariant basis vectors andthe unit basis vectors? Hint:Compare equation (10.11) and our discussion of it.

Point masses 1 at (1, 1, 1) and at (-1, 1, 1).

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

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

Write the transformation equations to show that $\mathbf{\nabla }\mathbf{\times }\mathit{V}$is a pseudo vector if Vis a vector. Hint:See equations (5.13), (6.2), and (6.3).