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Investigating the Meaning of a Function to the Right of the Del Operator
Let $F(x,y,z) \equiv f(x) g(y) h(z)$. In this previous post, $\vec{\nabla}F(x,y,z)$ was written in terms of spherical coordinates and unit vectors for a spherical coordinate system. The corresponding equation was found to be $ \vec{\nabla}F(x,y,z) = \vec{e}_r \frac{\partial F(x,y,z) }{\partial r} + \vec{e}_\theta \frac{1}{r} \frac{\partial F(x,y,z) }{\partial \theta} + \vec{e}_\phi \frac{1}{r\sin\theta} \frac{\partial F(x,y,z) }{\partial \phi} $. This is (6)…
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Writing Del in Terms of Spherical Coordinates
Objective The objective of this post is to investigate the validity of (6) in Reference 2. In this reference, (6) is del is written in terms of spherical coordinates and spherical unit vectors. Partial Derivatives Relating Cartesian Coordinates to Spherical Coordinates Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. In the this post, the following three equations were written. $ \frac{\partial…
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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} –…
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How a Multivariable Function Changes with Respect to the Cartesian Coordinates
Suppose $F(x,y,z) \equiv f(x) g(y) h(z)$. This post shows how to calculate a partial derivative of a multivariable function $F(x,y,z)$ with respect to each Cartesian coordinates. The resulting expressions are in spherical coordinates. In previous posts (see references [2], [3], and [4]), the following equations were written: $ \frac{\partial }{\partial r} F(x,y,z) = \bigg( ( \frac{\partial x}{\partial r}…