Relating Unit Vectors to a Jacobian Matrix

In this post, I relate coefficients of unit vectors to derivatives and to a Jacobian matrix that was used in a previous post.

Unit Vectors

Three unit vectors for a right-handed spherical coordinate system are

$  \vec{e}_r = \sin \theta \cos\phi \vec{e}_x + \sin \theta \sin\phi \vec{e}_y + \cos \theta \vec{e}_z $

$  \vec{e}_{\theta} =  \cos\theta \cos\phi \vec{e}_x + \cos\theta \sin\phi \vec{e}_y  -\sin\theta \vec{e}_z $

$\vec{e}_{\phi} = -\sin\phi \vec{e}_x + \cos\phi \vec{e}_{y}$.

In this coordinate system, a point is represented by $(r, \theta, \phi)$.

Derivatives

Recall the following nine derivatives from previous posts.

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

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

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

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

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

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

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

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

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

Notice that these derivatives are very similar to the coefficients of the unit vectors. In particular,

$ \vec{e}_r = \frac{dx(r)}{dr}\big|_{r^+} \vec{e}_x + \frac{dy(r)}{dr}\big|_{r^+} \vec{e}_y + \frac{dz(r)}{dr}\big|_{r^+} \vec{e}_z $

$ \vec{e}_{\theta} = \frac{1}{r} \frac{dx(\theta)}{d\theta} \big|_{\theta^+} \vec{e}_x + \frac{1}{r} \frac{d y(\theta)}{d \theta} \big|_{\theta^+}  \vec{e}_y  + \frac{1}{r} \frac{dz(\theta)}{d\theta} \big|_{\theta^+}  \vec{e}_z $

$\vec{e}_{\phi} = \frac{1}{r \sin\theta} \frac{dx(\phi)}{d\phi}\big|_{\phi^+}  \vec{e}_x + \frac{1}{r \sin\theta} \frac{dy(\phi)}{d\phi}\big|_{\phi^+}  \vec{e}_{y} + \frac{dz(\phi)}{d\phi}\big|_{\phi^+}  \vec{e}_z$.

The extra factors such as $\frac{1}{r}$ and $\frac{1}{r \sin\theta} $ account for change with respect to an arc length $s = r \alpha$ in which $\alpha$ is an arbitrary angle. The above equations can be represented with a matrix equation:

Next, recall this Jacobian matrix:

This Jacobian matrix is very similar to the first 3×3 matrix in the current post. The two matrices in this post are related by switching the row and columns. Switching the rows and columns of a matrix is also called taking the transpose of a matrix.

References

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

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