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Section 10.1 Gradient

As discussed in Section 9.8, 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{10.1.1} \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{10.1.2} \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, that is

\begin{equation} \grad{f} = \Partial{f}{x}\,\xhat + \Partial{f}{y}\,\yhat + \Partial{f}{z}\,\zhat .\tag{10.1.3} \end{equation}

Putting this all together, we have

\begin{equation} df = \grad{f} \cdot d\rr\label{Master}\tag{10.1.4} \end{equation}

which can in fact be taken as the geometric definition of the gradient, as further discussed in Section 10.2. We refer to (10.1.4) 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 \(d\rr\text{,}\) containing information about the particular change in position being made.

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