Coordinate Transformation & Jacobian Matrix

[Rasalhague]

Is the following correct, as far as it goes?

Suppose I have a vector space V and I'm making a transformation from one coordinate system, "the old system", with coordinates xⁱ, to another, "the new system", with coordinates yⁱ. Where i is an index that runs from 1 to n.

Let e_i denote the coordinate basis for the old system, and e'_i the coordinate basis for the new system.

I can define matrices B_L and B_R (where subscript L and R stand for "left" and "right") such that

$$B_L \begin{bmatrix} \vdots \\ \textbf{e}_i \\ \vdots \end{bmatrix} = \begin{bmatrix} \vdots \\ \textbf{e}'_i \\ \vdots \end{bmatrix}$$

$$\begin{bmatrix} \cdots & \textbf{e}_i & \cdots \end{bmatrix} B_R = \begin{bmatrix} \cdots & \textbf{e}'_i & \cdots \end{bmatrix}$$

and likewise matrices C_L and C_R, replacing the basis vectors in the above definitions with components of vectors in (the underlying set of) V.

And

$$C_L = \left ( C_R \right )^T = \begin{bmatrix} \frac{\partial y^1}{\partial x^1} & \cdots & \frac{\partial y^1}{\partial x^n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y^n}{\partial x^1} & \cdots & \frac{\partial y^n}{\partial x^n} \end{bmatrix}$$

and

$$\left ( C_L \right )^{-1} = B_L = \left ( B_R \right )^T = \begin{bmatrix} \frac{\partial x^1}{\partial y^1} & \cdots & \frac{\partial x^1}{\partial y^n} \\ \vdots & \ddots & \vdots \\ \frac{\partial x^n}{\partial y^1} & \cdots & \frac{\partial x^n}{\partial y^n} \end{bmatrix}.$$

And some people (e.g. Wolfram Mathworld, Berkley & Blanchard: Calculus) define the Jacobian matrix of this transformation as

$$J \equiv C_L \equiv \frac{\partial \left ( y^1,...,y^n \right )}{\partial \left ( x^1,...,x^n \right )}$$

while others (e.g. Snider & Davis: Vector Analysis) define it as

$$J \equiv \left ( C_L \right )^{-1} \equiv \frac{\partial \left ( x^1,...,x^n \right )}{\partial \left ( y^1,...,y^n \right )}.$$

 

[HallsofIvy]

They are really the same thing. Your C_L transforms from the "x_i" coordinates system to the "y_j" coordinate system while C_L^-1, of course, goes the other way, transforming from the "y_j" coordinate system to the "x_i" coordinate system.

 

[Rasalhague]

I don't understand how they're "really the same thing". For a given coordinate transformation, won't C_L generally be a different matrix from its inverse B_L? Also, changing a subscript on one of these matrices from L to R or vice versa transposes it, and in general a matrix is not the same thing as its transpose.

Experimenting with the transformation from 2d Cartesian to plane polar coordinates confirms that using the wrong matrices, or the right ones in the wrong order, gives the wrong answer. In fact in this thread, I did make a mistake (see #4), and if I'd done the multiplication correctly it wouldn't have given the required answer.

I'm thinking if they were literally the same, there'd be no need for Griffel's "Warning. There are two ways to define the change matrix. In our definition, the columns are the B'-components of the B vectors. Some authors define it with B and B' interchanged, giving a matrix which is the inverse of ours. The two versions of the theory look slightly different, but they are equivalent. It does not matter which version is used, provided itis used consistently. Using formulas from one version in a calculation from the other version will give the wrong answers" (Linear Algebra and its Applications, Vol. 2, p. 11).

But maybe you only meant that they're the same sort of thing, or that if we swapped the labels "old" and "new", the same matrices would be playing opposite roles.

 

[Landau]

But maybe you only meant that they're the same sort of thing, or that if we swapped the labels "old" and "new", the same matrices would be playing opposite roles.

Yes, this is correct.