Chapter 10: Tensor Analysis

$\begin{array}{l}\mathbf{x}\mathbf{=}\mathbf{uv}\\ \mathbf{y}\mathbf{=}\mathbf{u}\sqrt{\mathbf{1}\mathbf{-}{\mathbf{v}}^{\mathbf{2}}}\end{array}$

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions.In cylindrical coordinates$\mathbf{\nabla }\mathbf{.}{\mathit{e}}_{\mathbf{r}}\mathbf{,}\mathbf{\nabla }\mathbf{.}{\mathit{e}}_{\mathbf{\theta }}\mathbf{,}\mathbf{\nabla }\mathbf{\times }{\mathit{e}}_{\mathbf{r}}\mathbf{,}\mathbf{\nabla }\mathbf{\times }{\mathit{e}}_{\mathbf{\theta }}\mathbf{.}$ .

Continue Problem 8.15 to find the g^ijmatrix and the contravariant basis vectors. Check your result by solving the given equations for u and v in terms of x and y, and finding the contravariant basis vectors using Problem 12. On your Problem 8.15 sketches of the lines u=const. and v=const., also sketch the covariant and contravariant basis vectors. Observe that the covariant basis vectors lie along the lines u=const. and v=const. and the contravariant basis vectors lie along the normal to these lines.

In equation (5.16), show that if ${\mathbf{T}}_{\mathbf{jk}}$is a tensor (that is, not a pseudotensor), then ${\mathbf{V}}_{\mathbf{i}}$is a pseudovector (axial vector). Also show that if ${\mathbf{V}}_{\mathbf{i}}$is a pseudotensor, then ${\mathbf{T}}_{\mathbf{jk}}$is a vector (true or polar vector). You know that if role="math" localid="1659251751142" ${\mathbf{V}}_{\mathbf{i}}$is a cross product of polar vectors, then it is a pseudovector. Is its dual ${\mathbf{T}}_{\mathbf{jk}}$a tensor or a pseudotensor?

Repeat Problems 8.15and 10.16above for the (u,v)coordinate system if x=2u-v , y=u-2v.

Using (10.19), show that aⁱ a_j =𝛿 ⁱ _j.

Use equations (9.2), (9.8), and (9.11) to evaluate the following expressions. In spherical coordinates $\mathbf{\nabla }\mathbf{\times }\mathbf{\left(}\mathit{r}{\mathit{e}}_{\mathbf{\theta }}\mathbf{\right)}\mathbf{,}\mathbf{\nabla }\mathbf{\left(}\mathit{r}\mathit{c}\mathit{o}\mathit{s}\mathit{\theta }\mathbf{\right)}\mathbf{,}\mathbf{\nabla }\mathbf{.}\mathit{r}$.

Show that the transformation equation for a ${\mathbf{2}}^{\mathbf{n}\mathbf{d}}$-rank Cartesian tensor is equivalent to a similarity transformation. Warning hint: Note that the matrix C in Chapter 3 , Section 11 , is the inverse of the matrix A we are using in Chapter 10 (compare$\mathbf{}\mathbf{r}\mathbf{\text{'}}\mathbf{=}\mathbf{Ar}\mathbf{}\mathbf{and}\mathbf{}\mathbf{r}\mathbf{=}\mathbf{Cr}\mathbf{\text{'}}$). Thus a similarity transformation of the matrix T with tensor components${\mathit{T}}_{\mathbf{i}\mathbf{j}}$ is$\mathbf{T}\mathbf{\text{'}}\mathbf{=}{\mathbf{ATA}}^{\mathbf{-}\mathbf{1}}$. Also, see “Tensors and Matrices” in Section 3 and remember that A is orthogonal.

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.

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).