Figure 5.5 Basis vectors and one-forms for polar coordinates.

point. Moreover, the lengths of the bases are not constant. For example, from Eq. (5.23) we find

so that eθ increases in magnitude as we get further from the origin. The reason is that the basis vector eθ , having components (0,1) with respect to r and θ , has essentially a θ displacement of one unit, i.e. one radian. It must be longer to do this at large radii than at small. So we do not have a unit basis. It is easy to verify that

range of θ near the origin than farther away.

Metric tensor

The dot products above were all calculated by knowing the metric in Cartesian coordinates x, y:

A convenient way of displaying the components of g and at the same time showing the coordinates is the line element, which is the magnitude of an arbitrary ‘infinitesimal’

displacement dl:

equation are components of dl in polar coordinates, and ‘d’ simply means ‘infinitesimal ’. There is another way of deriving Eq. (5.33) which is instructive. Recall Eq. (3.26) in

Although this has a superficial resemblance to Eq. (5.33), it is different: it is an oper-

ator which, when supplied with the vector dl, the components of which are dr and dθ , gives Eq. (5.33). Unfortunately, the two expressions resemble each other rather too closely because of the confusing way notation has evolved in this subject. Most texts and research papers still use the ‘old-fashioned’ form in Eq. (5.33) for displaying the components of the metric, and we follow the same practice.

The metric has an inverse:

So, while (φ,r, φ,θ ) are components of a one-form, the vector gradient has components (φ,r, φ,θ /r2). Even though we are in Euclidean space, vectors generally have different components from their associated one-forms. Cartesian coordinates are the only coordinates in which the components are the same.