91Ó°ÊÓ

Parabolic cylinder.

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.

The square matrix in equation $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{3}\mathbf{)}$is called the Jacobian matrix J; the determinant of this matrix is the Jacobian $\mathbf{J}\mathbf{=}\mathbf{det}\mathbf{}\mathbf{J}$ which we used in Chapter 5 , Section 4 to find volume elements in multiple integrals. (Note that as in Chapter 3, J represents a matrix; J in italics is its determinant.) For the transformation to spherical coordinates in localid="1659266126385" $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{1}\mathbf{)}\mathbf{}$and $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{2}\mathbf{)}$ show that $\mathbf{J}\mathbf{=}\mathbf{det}\mathbf{}\mathbf{J}\mathbf{=}{\mathbf{r}}^{\mathbf{2}}\mathbf{sin}\mathbf{}\mathbf{θ}$ . Recall that the spherical coordinate volume element is ${\mathbf{r}}^{\mathbf{2}}\mathbf{sin}\mathbf{}\mathbf{θ»å°ù»åθ»åÏ•}$ . Hint: Find ${\mathbf{J}}^{\mathbf{T}}\mathbf{}\mathit{J}$ and note that $\mathbf{det}\mathbf{}\mathbf{\left(}{\mathbf{J}}^{\mathbf{T}}\mathbf{J}\mathbf{\right)}\mathbf{=}\mathbf{(}\mathbf{det}\mathbf{}\mathbf{J}{\mathbf{)}}^{\mathbf{2}}\mathbf{.}$

Do Problem 5 for the coordinate systems indicated in Problems 10 to 13. Parabolic.

Write and prove in tensor notation:

(a) Chapter 6, Problem 3.13.

(b) Chapter 6, Problem 3.14.

(c) Lagrange’s identity:$\left(A×B\right)\mathbf{·}\left(C×D\right)\mathbf{=}\left(A·C\right)\left(B·D\right)\mathbf{-}\left(A·D\right)\left(B·C\right)$.

(d), role="math" localid="1659335462905" $\mathbf{(}\mathbf{A}\mathbf{×}\mathbf{B}\mathbf{)}\mathbf{(}\mathbf{C}\mathbf{×}\mathbf{D}\mathbf{)}\mathbf{=}\mathbf{\left(}\mathbf{ABD}\mathbf{\right)}\mathbf{C}\mathbf{-}\mathbf{\left(}\mathbf{ABC}\mathbf{\right)}\mathbf{D}$where the symbol$\left(xyz\right)$ means the triple scalar product of the three vectors.

Show that in a general coordinate system with variables x₁, x₂, x₃, the contravariant basis vectors are given by

${\mathit{a}}^{\mathbf{i}}\mathbf{=}\mathbf{∇}{\mathit{x}}_{\mathbf{i}}\mathbf{=}\mathit{i}\frac{\mathbf{∂}{\mathbf{x}}_{\mathbf{i}}}{\mathbf{∂}\mathbf{x}}\mathbf{+}\mathit{j}\frac{\mathbf{∂}{\mathbf{x}}_{\mathbf{i}}}{\mathbf{∂}\mathbf{y}}\mathbf{+}\mathit{k}\frac{\mathbf{∂}{\mathbf{x}}_{\mathbf{i}}}{\mathbf{∂}\mathbf{z}}$

Hint:Write the gradient in terms of its covariant components and the basis

vectors to get$\mathbf{∇}\mathit{u}\mathbf{=}{\mathit{a}}^{\mathbf{j}}\frac{\mathbf{∂}\mathbf{u}}{\mathbf{∂}{\mathbf{x}}_{\mathbf{j}}}$and let$\mathit{u}\mathbf{=}{\mathit{x}}_{\mathbf{i}}$ .

Elliptical cylinder.

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

$\mathit{F}\mathbf{=}\mathit{q}\mathbf{(}\mathit{E}\mathbf{+}\mathit{v}\mathbf{×}\mathit{B}\mathbf{)}$

In equation$\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{13}\mathbf{)}\mathbf{,}$let the variables be rectangular coordinates x, y, z, and let ${\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{x}}_{\mathbf{2}}\mathbf{,}{\mathbf{x}}_{\mathbf{3}}$ , be general curvilinear coordinates, orthogonal or not (see end of Section 8 ). Show that ${\mathbf{J}}^{\mathbf{T}}\mathbf{J}$ is the $\mathbf{}{\mathbf{g}}_{\mathbf{ij}}\mathbf{}$matrix in$\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{13}\mathbf{)}$ [or in $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{16}\mathbf{)}$ for an orthogonal system]. Thus show that the volume element in a general coordinate system is $\mathbf{dV}\mathbf{=}\sqrt{\mathbf{g}}{\mathbf{dx}}_{\mathbf{1}}{\mathbf{dx}}_{\mathbf{2}}{\mathbf{dx}}_{\mathbf{3}}$ where$\mathbf{g}\mathbf{=}{\mathbf{detg}}_{\mathbf{ij}}$ , and that for an orthogonal system, this becomes [by $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{16}\mathbf{)}$ or $\mathbf{(}\mathbf{10}\mathbf{.}\mathbf{19}\mathbf{)}$ ], $\mathbf{dV}\mathbf{=}{\mathbf{h}}_{\mathbf{1}}{\mathbf{h}}_{\mathbf{2}}{\mathbf{h}}_{\mathbf{3}}{\mathbf{dx}}_{\mathbf{1}}{\mathbf{dx}}_{\mathbf{2}}{\mathbf{dx}}_{\mathbf{3}}$. Hint: To evaluate the products of partial derivatives in ${\mathbf{J}}^{\mathbf{T}}\mathbf{J}$, observe that the same expressions arise as in finding $\mathbf{}{\mathbf{ds}}^{\mathbf{2}}$ . In fact, from $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{11}\mathbf{)}$ and $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{12}\mathbf{)}$ , you can show that row i times column j in ${\mathbf{J}}^{\mathbf{T}}\mathbf{J}$ is just $\mathbf{}{\mathbf{a}}_{\mathbf{i}}\mathbf{.}{\mathbf{a}}_{\mathbf{j}}\mathbf{=}{\mathbf{g}}_{\mathbf{ij}}$ in equations $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{11}\mathbf{)}$ to $\mathbf{(}\mathbf{8}\mathbf{.}\mathbf{14}\mathbf{)}$ .