- #1
Rasalhague
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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 ei denote the coordinate basis for the old system, and e'i the coordinate basis for the new system.
I can define matrices BL and BR (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 matrices CL and CR, 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]
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 ei denote the coordinate basis for the old system, and e'i the coordinate basis for the new system.
I can define matrices BL and BR (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 matrices CL and CR, 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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