Chapter 10: Tensor Analysis

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 ${\mathbf{a}}_{\mathbf{r}}\mathbf{,}{\mathbf{a}}_{\mathbf{\theta }}\mathbf{,}{\mathbf{a}}_{\mathbf{\varphi }}$and the corresponding unit basis vectors${\mathbf{e}}_{\mathbf{r}}\mathbf{,}{\mathbf{e}}_{\mathbf{\theta }}\mathbf{,}{\mathbf{e}}_{\mathbf{\varphi }}$ . Write the ${\mathbf{g}}_{\mathbf{ij}}$matrix.

Verify equation (10.7). Hint: Use equations (2.4) to (2.6) and (2.10). For example,$\mathbf{\partial }{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{/}\mathbf{\partial }\mathit{z}\mathbf{=}\mathbf{\partial }\mathit{z}\mathbf{/}\mathbf{\partial }{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{=}{\mathit{n}}_{\mathbf{2}}\mathbf{=}{\mathit{a}}_{\mathbf{23}}$.

Verify equations(2.6).

As in (4.3) and (4.4), find the y and z components of (4.2) and the

other 6 components of the inertia tensor. Write the corresponding components

of the inertia tensor for a set of masses or an extended body as in (4.5).

Verify that (5.5) agrees with a Laplace development, say on the first row (Chapter 3, Section 3). Hints: You will find 6 terms corresponding to the 6 non-zero values of ${\mathit{\epsilon }}_{\mathbf{i}\mathbf{j}\mathbf{k}}$. First let;$\mathit{i}\mathbf{=}\mathbf{1}$ then j, k can be 2, 3 or 3, 2. These two terms give you ${\mathit{a}}_{\mathbf{11}}$times its cofactor. Next let$\mathit{i}\mathbf{=}\mathbf{\text{2}}$with$\mathit{j}\mathbf{,}\mathit{k}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{3}$and$\mathbf{3}\mathbf{,}\mathbf{1}$and show that you get times its cofactor. Finally let$\mathit{i}\mathbf{=}\mathbf{\text{3}}$. Watch all the signs carefully.

Verify Hints: In Figure ,$\mathbf{7}\mathbf{.1}$ consider the projection of the slanted face of area$\mathit{d}\mathit{S}$ onto the three unprimed coordinate planes. In each case, show that the projection angle is equal to an angle between the axis and one of the unprimed axes. Find the cosine of the angle from the matrix A in$\mathbf{(}\mathbf{2}\mathbf{.10}\mathbf{)}$ .

In cylindrical coordinates${\mathbf{\nabla }}^{\mathbf{2}}\mathit{r}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\left(\frac{1}{R}\right)\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathit{l}{\mathit{n}}_{\mathbf{r}}\mathbf{.}$

In spherical coordinates${\mathbf{\nabla }}^{\mathbf{2}}\mathit{r}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\mathbf{\left(}{\mathit{r}}^{\mathbf{2}}\mathbf{\right)}\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}\left(\frac{1}{r^2}\right)\mathbf{,}{\mathbf{\nabla }}^{\mathbf{2}}{\mathit{e}}^{\mathbf{i}\mathbf{k}\mathbf{r}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\theta }}$.

Let ${\mathit{e}}_{\mathbf{1}}\mathbf{,}{\mathit{e}}_{\mathbf{2}}\mathbf{,}{\mathit{e}}_{\mathbf{3}}$bea set of orthogonal unit vectors forming a right-handed system if taken in cyclic order. Show that the triple scalarproduct .${\mathit{e}}_{\mathbf{i}}\mathbf{.}\mathbf{(}{\mathit{e}}_{\mathbf{j}}\mathbf{\times }{\mathit{e}}_{\mathbf{k}}\mathbf{)}\mathbf{=}{\mathit{\epsilon }}_{\mathbf{i}\mathbf{j}\mathbf{k}}$

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.