Chapter 10: Tensor Analysis

P Derive the expression (9.11)for curl V in the following way. Show that $\mathbf{\nabla }{\mathbf{x}}_{\mathbf{1}}\mathbf{=}\frac{{\mathbf{e}}_{\mathbf{1}}}{{\mathbf{h}}_{\mathbf{1}}}$ and $\mathbf{\nabla }\mathbf{\times }\mathbf{(}\mathbf{\nabla }{\mathit{x}}_{\mathbf{1}}\mathbf{)}\mathbf{=}\mathbf{\nabla }\mathbf{\times }\mathbf{\left(}\frac{{\mathbf{e}}_{\mathbf{1}}}{{\mathbf{h}}_{\mathbf{1}}}\mathbf{\right)}\mathbf{=}\mathbf{0}$ . Write V in the form $\mathit{V}\mathbf{=}\frac{{\mathbf{e}}_{\mathbf{1}}}{{\mathbf{h}}_{\mathbf{1}}}\mathbf{\left(}{\mathit{h}}_{\mathbf{1}}{\mathit{V}}_{\mathbf{1}}\mathbf{\right)}\mathbf{+}\frac{{\mathbf{e}}^{\mathbf{2}}}{{\mathbf{h}}_{\mathbf{2}}}\left(h_2{V}_2\right)\mathbf{+}\frac{{\mathbf{e}}_{\mathbf{3}}}{{\mathbf{h}}_{\mathbf{3}}}\left(h_3{V}_3\right)$and use vector identities from Chapter 6 to complete the derivation.

Do Example 1 and Problem 3 if the transformation to a left-handed system is an inversion (see Problem 2).

Show that the sum of the squares of the direction cosines of a line through the origin is equal to 1 Hint: Let $\left(a,b,c\right)$be a point on the line at distance 1 from the origin. Write the direction cosines in terms of $\left(a,b,c\right)$.

Show that the fourth expression in (3.1) is equal to $\frac{\mathbf{\partial }\mathbf{u}}{\mathbf{\partial }\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{i}}}$. By equations (2.6) and (2.10) , show that $\frac{\mathbf{\partial }{\mathbf{x}}_{\mathbf{j}}}{\mathbf{\partial }\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{i}}}\mathbf{=}\mathit{a}$, so $\frac{\mathbf{\partial }\mathbf{u}}{\mathbf{\partial }\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{i}}}\mathbf{=}\frac{\mathbf{\partial }\mathbf{u}}{\mathbf{\partial }{\mathbf{x}}_{\mathbf{j}}}\frac{\mathbf{\partial }{\mathbf{x}}_{\mathbf{j}}}{\mathbf{\partial }\mathbf{x}{\mathbf{\text{'}}}_{\mathbf{i}}}\mathbf{=}{\mathit{a}}_{\mathbf{i}\mathbf{j}}\frac{\mathbf{\partial }\mathbf{u}}{\mathbf{\partial }{\mathbf{x}}_{\mathbf{j}}}$

Compare this with equation (2.12) to show that$\mathbf{\nabla }\mathit{u}$is a Cartesian vector. Hint: Watch the summation indices carefully and if it helps, put back the summation signs or write sums out in detail as in (3.1) until you get used to summation convention.

Write out the sums${\mathit{P}}_{\mathbf{i}\mathbf{j}}{\mathit{e}}_{\mathbf{j}}$ for each value of and compare the discussion of $\mathbf{(}\mathbf{1}\mathbf{.1}\mathbf{)}$.Hint: For example, if$\mathit{i}\mathbf{=}\mathbf{2}$ [or y in$\mathbf{(}\mathbf{1}\mathbf{.1}\mathbf{)}$ ], then the pressure across the face perpendicular to the${\mathit{x}}_{\mathbf{2}}$axis is $P_{21}{e}_1+P_{22}{e}_{22}+P_{23}{e}_3$, or, in the notation of (1.1), ${\mathit{P}}_{\mathbf{y}\mathbf{x}}\mathit{i}\mathbf{+}{\mathit{P}}_{\mathbf{y}\mathbf{y}}\mathit{j}\mathbf{+}{\mathit{P}}_{\mathbf{y}\mathbf{z}}\mathit{k}$.

Verify for a few representative cases that $\mathbf{(}\mathbf{5}\mathbf{.6}\mathbf{)}$gives the same results as a Laplace development. First note that if$\mathit{\alpha }\mathbf{,}\mathit{\beta }\mathbf{,}\mathit{\gamma }\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}$ , then $\mathbf{(}\mathbf{5}\mathbf{.6}\mathbf{)}$is just$\mathbf{(}\mathbf{5}\mathbf{.5}\mathbf{)}$ . Then try letting$\mathit{\alpha }\mathbf{,}\mathit{\beta }\mathbf{,}\mathit{\gamma }\mathbf{=}$ an even permutation of $\mathbf{1}\mathbf{,}\mathbf{2}\mathbf{,}\mathbf{3}$, and then try an odd permutation, to see that the signs work out correctly. Finally try a case when ${\mathit{\epsilon }}_{\mathbf{\alpha }\mathbf{\beta }\mathbf{\gamma }}\mathbf{=}\mathbf{0}$(that is when two of the indices are equal) to see that the right hand side of$\mathbf{(}\mathbf{5}\mathbf{.6}\mathbf{)}$ is zero because you are evaluating a determinant which has two identical rows.

Complete Example 4 to verify the rest of the components of the inertia tensor and the principal moments of inertia and principal axes. Verify that the three principal axes form an orthogonal triad.

Observe that a simpler way to find the velocity $\frac{\mathbf{ds}}{\mathbf{dt}}$in (8.10)is to divide the vectordsin (8.6)by. Complete the problem to find the acceleration in cylindrical coordinates.

In Chapter, Problem , you are asked to prove some identities among the Pauli spin matrices (called A, B, C, in that problem). Call the Pauli spin matrices ${\mathit{\sigma }}_{\mathbf{1}}\mathbf{,}{\mathit{\sigma }}_{\mathbf{2}}\mathbf{,}{\mathit{\sigma }}_{\mathbf{3}}$; then show that the identities can be written in the following summation forms:

$$\begin{gathered} {\mathit{\sigma }}_{\mathbf{k}}{\mathit{\sigma }}_{\mathbf{m}}\mathbf{=}\mathit{i}{\mathit{\epsilon }}_{\mathbf{k}\mathbf{m}\mathbf{n}}{\mathit{\sigma }}_{\mathbf{n}}\mathbf{+}{\mathit{\delta }}_{\mathbf{k}\mathbf{m}} \\ {\mathit{\sigma }}_{\mathbf{k}}{\mathit{\sigma }}_{\mathbf{m}}{\mathit{\epsilon }}_{\mathbf{k}\mathbf{m}\mathbf{n}}\mathbf{=}\mathbf{2}\mathit{i}{\mathit{\sigma }}_{\mathbf{n}} \end{gathered}$$

Show that ${\mathit{\delta }}_{\mathbf{i}\mathbf{j}}{\mathit{ϵ}}_{\mathbf{k}\mathbf{l}\mathbf{m}}$is an isotropic tensor of rank 5. Hint: Combine equations (5.4) and (5.7).