Section 6.1 The Vector Differential \(d\rr\)
The position vector
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 6.1.1, 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
as shown in Figure 6.1.2, 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.
What is the infinitesimal distance \(ds\) between nearby points? Just the length of \(d\rr\text{.}\) We have
and squaring both sides leads to
which is just the infinitesimal Pythagorean Theorem, the two-dimensional version of which is shown in Figure 6.1.2. 1