As discussed in § {The Multivariable Differential}, 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 \end{equation} Each term is a product of two factors, labeled by $x$, $y$, and $z$. 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) \end{equation} The last factor is just $d\rr$, and you may recognize the first factor as the gradient of $f$ written in rectangular coordinates, that is \begin{equation} \grad{f} = \Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat + \Partial{f}{z}\,\zhat \end{equation} Putting this all together, we have \begin{equation} df = \grad{f} \cdot d\rr \label{Master} \end{equation} which can in fact be taken as the geometric definition of the gradient, as further discussed in § {The Geometry of Gradient}. We refer to ($\ref{Master}$) as the Master Formula, because it contains all of the information needed to determine the gradient, and does so without relying on a particular coordinate system.

Recall that $df$ represents the infinitesimal change in $f$ when moving to a “nearby” point. What information do you need in order to know how $f$ changes? You must know something about how $f$ behaves, where you started, and which way you went. The master formula organizes this information into two geometrically different pieces, namely the gradient, containing generic information about how $f$ changes, and the vector differential $d\rr$, containing information about the particular change in position being made.