Suppose that we wish to discuss from the physical point of view how a field of force varies from point to point. If polar coordinates are being used, a change of the $r$-component does not necessarily indicate a want of uniformity in the field of force; it is at least partly attributable to the inclination between the $r$-directions at different points. Similarly when rotating axes are used, the rate of change of momentum~$h$ is given not by $dh_{1}/dt$, etc., but by \[ dh_{1}/dt - \omega_{3}/h_{2} + \omega_{2}/h_{3}, \text{ etc.} \Tag{(33.1)} \] The momentum may be constant even when the time-derivatives of its components are not zero. We must recognise then that the change of a physical entity is usually regarded as something distinct from the change of the mathematical components into which we resolve it. In the elementary theory a definition of the former change is obtained by identifying it with the change of the components in unaccelerated rectangular coordinates; but this is of no avail in the general case because space-time may be of a kind for which no such coordinates exist. Can we still preserve this notion of a \emph{physical} rate of change in the general case? Our attention is directed to the rate of change of a physical entity because of its importance in the laws of physics, e.g.\ force is the time-rate of change \PageSep{69} of momentum, or the space-rate of change of potential; therefore the rate of change should be expressed by a tensor of some kind in order that it may enter into the general physical laws. Further in order to agree with the customary definition in elementary cases, it must reduce to the rate of change of the rectangular components when the coordinates are Galilean. Both conditions are fulfilled if we define the physical rate of change of the tensor by its covariant derivative. The covariant derivative~$A_{\mu\nu}$ consists of the term $\dd A_{\mu}/dx_{\nu}$, giving the apparent gradient, from which is subtracted the ``spurious change'' $\{\mu\nu, \alpha\}\, A_{\alpha}$ attributable to the curvilinearity of the coordinate-system. When Cartesian coordinates (rectangular or oblique) are used, the $3$-index symbols vanish and there is, as we should expect, no spurious change. For the present we shall call~$A_{\mu\nu}$ the rate of \emph{absolute change} of the vector~$A_{\mu}$. \index{Absolute change}% Consider an elementary mesh in the plane of~$x_{\nu} x_{\sigma}$, the corners being at \[ A(x_{\nu}, x_{\sigma}),\quad B(x_{\nu} + dx_{\nu}, x_{\sigma}),\quad C(x_{\nu} + dx_{\nu}, x_{\sigma} + dx_{\sigma}),\quad D(x_{\nu}, x_{\sigma} + dx_{\sigma}). \] Let us calculate the whole absolute change of the vector-field~$A_{\mu}$ as we pass round the circuit $ABCDA$.