The Jacobian Matrix

Using conclusions from previous posts, the following nine derivatives have been determined.

$\frac{dx(r)}{dr}\big|_{r^+}=\sin\theta\cos\phi$

$\frac{dy(r)}{dr}\big|_{r^+} = \sin\theta\sin\phi$

$\frac{dz(r)}{dr}\big|_{r^+} = \cos\theta$

$\frac{dx(\phi)}{d\phi}\big|_{\phi^+} = -r\sin\theta\sin\phi$

$\frac{dy(\phi)}{d\phi}\big|_{\phi^+} = r \sin\theta \cos\phi$

$\frac{dz(\phi)}{d\phi}\big|_{\phi^+} = 0$

$\frac{dx(\theta)}{d\theta} \big|_{\theta^+} = r\cos\theta \cos\phi$

$\frac{d y(\theta)}{d \theta} \big|_{\theta^+} = r \cos\theta \sin\phi $

$\frac{dz(\theta)}{d\theta} \big|_{\theta^+} = -r \sin\theta$


Next, recall the following result from this post.

This matrix equation can be rewritten by substituting the nine derivatives listed at the beginning of the current post. Replacing each $a$ with the corresponding independent variable, the result is:

This matrix equation includes nine equations relating dummy variables in Cartesian coordinates to dummy variables in spherical coordinates.

According to reference 1, the matrix on the right–without the dummy variables–is exactly the Jacobian matrix, if $r \rightarrow \rho$, $\theta \rightarrow \varphi$, and $\phi \rightarrow \theta$. The Jacobian matrix from reference 1 is included below.

That said, with division the previous matrix equation can be simply written as follows.

The matrix on the right is the Jacobian matrix for converting from cartesian coordinates to spherical coordinates. In retrospect, I could have directly formed the matrix on the right by using all nine derivatives at the beginning of the current post, but at least the above approach shows how the Jacobian matrix is related to the dummy variables that appear in integrals and derivatives.

References:

[1] https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant

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