Changes

::<math>\begin{bmatrix} A_u \\ A_v \\ A_w \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial u}{\partial x} & \displaystyle\frac{\partial u}{\partial y} & \displaystyle\frac{\partial u}{\partial z} \\ \\ \displaystyle\frac{\partial v}{\partial x} & \displaystyle\frac{\partial v}{\partial y} & \displaystyle\frac{\partial v}{\partial z} \\ \\ \displaystyle\frac{\partial w}{\partial x} & \displaystyle\frac{\partial w}{\partial y} & \displaystyle\frac{\partial w}{\partial z}\end{bmatrix} \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}</math>

::<math>\begin{bmatrix} A_u \\ A_v \\ A_w \end{bmatrix} = \begin{bmatrix} \displaystyle\frac{\partial u}{\partial x} & \displaystyle\frac{\partial u}{\partial y} & \displaystyle\frac{\partial u}{\partial z} \\ \\ \displaystyle\frac{\partial v}{\partial x} & \displaystyle\frac{\partial v}{\partial y} & \displaystyle\frac{\partial v}{\partial z} \\ \\ \displaystyle\frac{\partial w}{\partial x} & \displaystyle\frac{\partial w}{\partial y} & \displaystyle\frac{\partial w}{\partial z}\end{bmatrix} \begin{bmatrix} A_x \\ A_y \\ A_z \end{bmatrix}</math>

In other words, the transformation of vector components is just a matrix multiplication by the Jacobian matrix

In other words, the transformation of vector components from x/y/z to u/v/w is just a matrix multiplication by the Jacobian matrix

::<math>\begin{bmatrix}J(u, v, w / x, y, z)\end{bmatrix}</math>

::<math>\begin{bmatrix}J(u, v, w / x, y, z)\end{bmatrix}</math>

We make a simplification to avoid having our notation get completely out of control.  In keeping with our goal of expressing everything just in terms of the new coordinate system:

We make a simplification to avoid having our notation get completely out of control.  In keeping with our goal of expressing everything just in terms of the new coordinate system:

We can combine the two Jacobians into one matrix, and then use that whenever we need a dot product.  Let:

We can combine the two Jacobians into one matrix, and then use that whenever we need a dot product.  Let:

::<math>[g]_{jk} = \sum_i [J]_{ij} [J]_{ij} = \sum_i [J]^t_{ji} [J]_{ij}</math>

::<math>[g]_{jk} = \sum_i [J]_{ij} [J]_{ik} = \sum_i [J]^t_{ji} [J]_{ik}</math>

That is, [g] is just the matrix product of the transpose of [J] and [J].

That is, [g] is just the matrix product of the transpose of [J] and [J] itself.

::<math>[g]_{jk} = ([J]^t [J])_{jk}\,</math>

::<math>[g]_{jk} = ([J]^t [J])_{jk}\ \ \ or\ \ \ [g] = [J]^t [J]\,</math>

Then:

Then:

::<math>\vec{A} \cdot \vec{B} = \sum_{jk} [g]_{jk} A_j B_k</math>

::<math>\vec{A} \cdot \vec{B} = \sum_{jk} [g]_{jk} A_j B_k</math>

===The Divergence in Arbitrary Coordinates===

===The Divergence in Arbitrary Coordinates===

Here's the definition of the divergence operator in arbitrary curvilinear coordinates:

Here's the definition of the divergence operator in arbitrary curvilinear coordinates:

::<math>\nabla \cdot \vec{A} = \frac{1}{\sqrt{g}} \sum_i \frac{\partial}{\partial x_i}\ (\sqrt{g}\ A_i) = \sum_i \frac{\partial A_i}{\partial x_i} + \frac{1}{\sqrt{g}}\ \sum_i \frac{\partial \sqrt{g}}{\partial x_i}\  A_i</math>

::<math>\nabla \cdot \vec{A} = \frac{1}{\sqrt{g}} \sum_i \frac{\partial}{\partial x_i}\ (\sqrt{g}\ A_i) = \sum_i \frac{\partial A_i}{\partial x_i} + \frac{1}{\sqrt{g}}\ \sum_i \frac{\partial \sqrt{g}}{\partial x_i}\  A_i</math>

In practice, it's not as forbidding as it looks, for vector fields that are well suited to the particular coordinate system.

In practice, it's not as forbidding as it looks, for vector fields that are well suited to the particular coordinate system.

Then the curl is:

Then the curl is:

::<math>\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ B_u & B_v & B_w \end{vmatrix}</math>

::<math>\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ B_u & B_v & B_w \end{vmatrix}</math>

::Or, if the coordinate system is orthogonal:

::Or, if the coordinate system is orthogonal:

::::<math>\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ g_{11}A_u & g_{22}A_v & g_{33}A_w \end{vmatrix}</math>

::::<math>\nabla \times \vec{A} = \frac{1}{\sqrt{g}} \begin{vmatrix} \hat{u} & \hat{v} & \hat{w} \\ \\ \displaystyle\frac{\partial}{\partial u} & \displaystyle\frac{\partial}{\partial v} & \displaystyle\frac{\partial}{\partial v} \\ \\ g_{11}A_u & g_{22}A_v & g_{33}A_w \end{vmatrix}</math>

::::***** Need to put in some examples, etc.  I have them worked out, but typing them in is tedious.

::::***** Need to put in some examples, etc.  I have them worked out, but typing them in is tedious.

=Problems=

=Problems=