
Writing Unit Vectors for a Cartesian Coordinate System in Terms of Unit Vectors for a Spherical Coordinate System
Objective of this Post The objective of this post is to form (4) in this reference. Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the previous post, the following three equations were written. $ \frac{\partial F(x,y,z)}{\partial x} = \cos\phi \sin\theta \frac{\partial F}{\partial r} + \frac{\cos \phi \cos\theta}{r} \frac{\partial F}{\partial \theta} –…

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 righthanded 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…