Section 9.2 The Gradient in Rectangular Coordinates
As discussed in Section 6.9, the chain rule for a function of several variables, written in terms of differentials, takes the form:
\begin{equation}
df = \Partial{f}{x}\,dx + \Partial{f}{y}\,dy + \Partial{f}{z}\,dz .\tag{9.2.1}
\end{equation}
Each term is a product of two factors, labeled by \(x\text{,}\) \(y\text{,}\) and \(z\text{.}\) This looks like a dot product. Separating out the pieces, we have
\begin{equation}
df =
\left(
\Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat + \Partial{f}{z}\,\zhat
\right)
\cdot (dx\,\xhat + dy\,\yhat + dz\,\zhat) .\tag{9.2.2}
\end{equation}
The last factor is just \(d\rr\text{,}\) and you may recognize the first factor as the gradient of \(f\) written in rectangular coordinates. Putting this all together, this algebraic observation leads us back to (9.1.2), as further discussed in Section 9.1.
With our new interpretation of (9.1.2) as the definition of the gradient, the argument above shows how to determine the formula for the gradient in rectangular coordinates, namely
\begin{equation}
\grad{f}
= \Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat + \Partial{f}{z}\,\zhat .\tag{9.2.3}
\end{equation}