Ch.2 Operations with Nabla Operator

Table of Contents

After introducing the mathematical definitions of gradient, divergence, and curl in the last chapter. Let us look at some useful identities that will come in handy when you are actually computing the operations. We will shall also examine two cases where the combination of two operations give a noteworthy result. 

Operators

We ended the last chapter by introducing the \(\nabla\) operator. What is exactly an operator?  For our purpose, it is a mean to extract information from a system. This definition both apply in the mathematical sense, as well as in the physical sense when you do quantum mechanics. There are several conventions associated with the \(\nabla\) operator.

1. It is non-commutable.

We know in algebra, \(ab=ba\), where \(a,b\in\mathbb{R}\). But for operators, it doesn’t apply:

\begin{align} \div \phi &=\pdv{\phi}{x}+\pdv{\phi}{y}+\pdv{\phi}{z} \\ \phi\nabla &=\phi\pdv{x}+\phi\pdv{y}+\phi\pdv{z}\\ \div \phi &\neq \phi \nabla\end{align}

where

eq

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