The Vector Differential d\rr

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Section 8.1 The Vector Differential \(d\rr\)

Figure 8.1. The infinitesimal displacement vector \(d\rr\) along a curve, shown in an “infinite magnifying glass”. In this and subsequent figures, artistic license has been taken in the overall scale and the location of the origin in order to make a pedagogical point.

The position vector

\begin{equation} \rr=x\,\xhat+y\,\yhat+z\,\zhat\tag{8.1.1} \end{equation}

describes the location of the point \((x,y,z)\) in rectangular coordinates, and is usually thought of as pointing from the origin to that point. It is instructive to draw a picture of the small change \(\Delta\rr=\Delta x\,\xhat + \Delta y\,\yhat + \Delta z\,\zhat\) in the position vector between nearby points. Try it! This picture is so useful that we will go one step further, and consider an infinitesimal change in position. Instead of \(\Delta\rr\text{,}\) we will write \(d\rr\) for the vector between two points which are infinitesimally close together. This is illustrated in Figure 8.1, which shows a view of the curve through an “infinite magnifying glass”.

Like any vector, \(d\rr\) can be expanded with respect to \(\xhat\text{,}\) \(\yhat\text{,}\) \(\zhat\text{;}\) the components of \(d\rr\) are just the infinitesimal changes \(dx\text{,}\) \(dy\text{,}\) \(dz\text{,}\) in the \(x\text{,}\) \(y\text{,}\) and \(z\) directions, respectively, that is

\begin{equation} d\rr = dx\,\xhat + dy\,\yhat + dz\,\zhat\tag{8.1.2} \end{equation}

as shown in Figure 8.2, The geometric notion of \(d\rr\) as an infinitesimal vector displacement will be a unifying theme to help us in visualizing the geometry of all of vector calculus.

Figure 8.2. The first figure shows the rectangular components of the vector differential \(d\rr\) in two dimensions, while the second figure shows the infinitesimal version of the Pythagorean Theorem.

Figure 8.2. The first figure shows the rectangular components of the vector differential \(d\rr\) in two dimensions, while the second figure shows the infinitesimal version of the Pythagorean Theorem.

What is the infinitesimal distance \(ds\) between nearby points? Just the length of \(d\rr\text{.}\) We have

\begin{equation} ds = |d\rr| = \sqrt{d\rr\cdot d\rr} = \sqrt{dx^2 + dy^2 + dz^2}\tag{8.1.3} \end{equation}

and squaring both sides leads to

\begin{equation} ds^2 = |d\rr|^2 = d\rr\cdot d\rr = dx^2 + dy^2 + dz^2\tag{8.1.4} \end{equation}

which is just the infinitesimal Pythagorean Theorem, the two-dimensional version of which is shown in Figure 8.2. ¹

When dealing with infinitesimals, we prefer to avoid second-order errors by anchoring all vectors to the same point.