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I am reading R.A. Sharipov'sQuick Introduction to Tensor Analysis, and I am stuck on the following issue, on pages 38-39. The text is freely available here: http://arxiv.org/abs/math/0403252. If my understanding is correct, then the Jacobi matrices for the direct and inverse coordinates transformation are inverse of each other (when computed in the same point using the same frame of reference, of course). I want to apply the concept to spherical coordinates.

The spherical coordinates transformation can be defined as follows:

[tex]

\begin{cases}

x^1 & = x & = \rho \cos \theta \sin \phi \\

x^2 & = y & = \rho \sin \theta \sin \phi \\

x^3 & = z & = \rho \cos \phi

\end{cases}

[/tex]

and its inverse is:

[tex]

\begin{cases}

y^1 & = \rho & = √x^2+y^2+z^2 \\

y^2 & = \theta & = \arctan(y/x) \\

y^3 & = \phi & = \arccos(z/r)

\end{cases}

[/tex]

The Jacobi matrices for the two transformations are defined respectively as:

[tex]

\begin{cases}

S^i_j & =∂x^i/∂y^j \\

T^i_j & =∂y^i/∂x^j.

\end{cases}

[/tex]

Switching to matrix notation: if those matrices are inverse to each other, then I should get ##ST=I## where ##I## is the identity matrix.

By applying the definitions, I get the following matrices:

[tex]

S=

\begin{pmatrix}

\cos\theta\sin\phi & \sin\theta\sin\phi & \cos\phi \\

-\rho\sin\theta\sin\phi & \rho\cos\theta\sin\phi & 0 \\

\rho\cos\theta\cos\phi & \rho\sin\theta\cos\phi & -\rho\sin\phi

\end{pmatrix}

[/tex]

and

[tex]

T=

\begin{pmatrix}

x/\rho & y/\rho & z/\rho \\

-y/(r^2-z^2) & x/(r^2-z^2) & 0 \\

-xz/r^2\sqrt{r^2-z^2} & -yz/r^2\sqrt{r^2-z^2} & \sqrt{r^2-z^2}/\rho^2

\end{pmatrix}

=

\begin{pmatrix}

\cos\theta\sin\phi & \sin\theta\sin\phi & \cos\phi \\

-\sin\theta/\rho\sin\phi & \cos\theta/\rho\sin\phi & 0 \\

-\cos\theta\cos\phi/\rho & -\sin\theta\cos\phi/\rho & \sin\phi/\rho

\end{pmatrix}.

[/tex]

Each row is the gradient of a component of the vector transformation. Clearly, ##ST\neq I##. However, ##ST^t=I##. Why? What's wrong with my reasoning above?

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# Inverse Jacobi Matrix in Spherical Coordinates

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