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Suppose I have a vector spaceVand I'm making a transformation from one coordinate system, "the old system", with coordinatesx^{i}, to another, "the new system", with coordinatesy^{i}. Whereiis an index that runs from 1 to n.

Lete_{i}denote the coordinate basis for the old system, ande'_{i}the coordinate basis for the new system.

I can define matricesB_{L}andB_{R}(where subscript L and R stand for "left" and "right") such that

[tex]B_L \begin{bmatrix} \vdots \\ \textbf{e}_i \\ \vdots \end{bmatrix} = \begin{bmatrix} \vdots \\ \textbf{e}'_i \\ \vdots \end{bmatrix}[/tex]

[tex]\begin{bmatrix} \cdots & \textbf{e}_i & \cdots \end{bmatrix} B_R = \begin{bmatrix} \cdots & \textbf{e}'_i & \cdots \end{bmatrix}[/tex]

and likewise matricesC_{L}andC_{R}, replacing the basis vectors in the above definitions with components of vectors in (the underlying set of)V.

And

[tex]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}[/tex]

and

[tex]\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}.[/tex]

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

[tex]J \equiv C_L \equiv \frac{\partial \left ( y^1,...,y^n \right )}{\partial \left ( x^1,...,x^n \right )}[/tex]

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

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

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# Jacobian matrix

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