SkipCaptcha, Automoderated users, edit
3,266
edits
Changes
Jump to navigation
Jump to search
Line 285:
Line 285: − + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
→More About Coordinate Systems: That's enough for today.
We make another 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 another 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:
*The matrix that we have been calling [J(x, y, z / u, v, w)] we will call just [J].
*The matrix that we have been calling [J(x, y, z / u, v, w)] we will call just [J].
*The matrix that goes the other way, formerly called [J(u, v, w / x, y, z)] we will call <math>[J]^{-1}</math>. It's just the inverse of the other matrix.
*The matrix that goes the other way, formerly called [J(u, v, w / x, y, z)] we will call <math>[J]^{-1}</math>. It's just the inverse of the first matrix.
We will call [J] the "Jacobian of the coordinate system".
We will call [J] the "Jacobian of the coordinate system".
::The astute reader will notice that this is not a good definition. [J] depends on the choice of the "reference" Cartesian system, so we're not justified in calling it just "the Jacobian". But it turns out that it won't make any difference. We will always get the same answers, no matter what reference system we used. See the exercises.
::The astute reader will notice that this is not a good definition. [J] depends on the choice of the "reference" Cartesian system, so we're not justified in calling it just "the Jacobian". But it turns out that it won't make any difference. We will always get the same answers, no matter what reference system we used. See the exercises.
We can now calculate the dot product of <math>\vec{A}</math> and <math>\vec{B}</math>, from their components and the Jacobian. We know that
::<math>\vec{A} \cdot \vec{B} = \sum_i \tilde{A_i} \tilde{B_i}</math>
in terms of the components in the reference system. We also have the transformation rule in terms of the Jacobian matrix.
::<math>\tilde{A_i} = \sum_j [J(x, y, z / u, v, w)]_{ij} A_j</math>
This is just matrix multiplication. The summation is from 1 to 3, or whatever the dimension is.
By our new convention, this is just the Jacobian matrix [J], so
::<math>\tilde{A_i} = \sum_j [J]_{ij} A_j</math>
::<math>\tilde{B_i} = \sum_k [J]_{ik} B_k</math>
::<math>\vec{A} \cdot \vec{B} = \sum_i \tilde{A_i} \tilde{B_i} = \sum_{ijk} [J]_{ij} [J]_{ik} A_j B_k</math>
As promised, no more tildes!
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>
That is, [g] is just the matrix product of the transpose of [J] and [J].
::<math>[g]_{jk} = ([J]^t [J])_{jk}\,</math>
Then:
::<math>\vec{A} \cdot \vec{B} = \sum_{jk} [g]_{jk} A_j B_k</math>
This is the rule for the dot product in general coordinates. The matrix [g] is called the ''metric tensor'' or just the ''metric''.
For polar coordinates in 2 dimensions:
::<math>[J] = \begin{bmatrix} \cos \theta & - r \sin \theta \\ \sin \theta & r \cos \theta \end{bmatrix}</math>
::<math>[g] = \begin{bmatrix} 1 & 0 \\ 0 & r^2 \end{bmatrix}</math>
So, in polar coordinates:
::<math>\vec{A} \cdot \vec{B} = A_1 B_1 + r^2 A_2 B_2</math>
or, going back to giving the coordinates names:
::<math>\vec{A} \cdot \vec{B} = A_r B_r + r^2 A_\theta B_\theta</math>
For spherical coordinates in 3 dimensions:
::<math>[J] = \begin{bmatrix} \sin \theta \cos \phi & r \cos \theta \cos \phi & - r \sin \theta \sin \phi \\ \sin \theta \sin \phi & r \cos \theta \sin \phi & r \sin \theta \cos \phi \\ \cos \theta & - r \sin \theta & 0 \end{bmatrix}</math>
::<math>[g] = \begin{bmatrix} 1 & 0 & 0 \\ 0 & r^2 & 0 \\ 0 & 0 & r^2 \sin^2 \theta \end{bmatrix}</math>
The metric is always a symmetric matrix. (Exercise: Prove this. Use the definition in terms of <math>[J]</math> and <math>[J]^{-1}</math>.) For all coordinate systems that have the coordinate lines crossing at right angles everywhere (which includes all the common systems), the metric is a ''diagonal matrix''—the only nonzero items are on the main diagonal.
===The Cross Product in Arbitrary Coordinates===
===The Cross Product in Arbitrary Coordinates===