The differentials $(dx_{1}, dx_{2}, dx_{3}, dx_{4})$ are transformed according to the equations~\Eq{(15.2)}, viz.\ \[ dx_{1}' = \frac{\dd x_{1}'}{\dd x_{1}}\, dx_{1} + \frac{\dd x_{1}'}{\dd x_{2}}\, dx_{2} + \frac{\dd x_{1}'}{\dd x_{3}}\, dx_{3} + \frac{\dd x_{1}'}{\dd x_{4}}\, dx_{4}; \text{ etc.,} \] which may be written shortly \[ dx_{\mu}' = \sum_{\alpha=1}^{4} \frac{\dd x_{\mu}'}{\dd x_{\alpha}}\, dx_{\alpha}, \] four equations being obtained by taking $\mu = 1$, $2$, $3$, $4$, successively. Any set of four quantities transformed according to this law is called a \emph{contravariant vector}. Thus if $(A^{1}, A^{2}, A^{3}, A^{4})$ becomes $(A'^{1}, A'^{2}, A'^{3}, A'^{4})$ in \index{Vector}% the new coordinate-system, where \[ A'^{\mu} = \sum_{\alpha=1}^{4} \frac{\dd x_{\mu}'}{\dd x_{\alpha}}\, A^{\alpha}, \Tag{(19.1)} \] then $(A^{1}, A^{2}, A^{3}, A^{4})$, denoted briefly as~$A^{\mu}$, is a contravariant vector. The upper position of the suffix (which is, of course, not an exponent) is reserved to indicate contravariant vectors. If $\phi$~is an invariant function of position, i.e.\ if it has a fixed value at each point independent of the coordinate-system employed, the four quantities \[ \left(\frac{\dd\phi}{\dd x_{1}}, \frac{\dd\phi}{\dd x_{2}}, \frac{\dd\phi}{\dd x_{3}}, \frac{\dd\phi}{\dd x_{4}}\right) \] are transformed according to the equations \[ \frac{\dd\phi}{\dd x_{1}'} = \frac{\dd x_{1}}{\dd x_{1}'}\, \frac{\dd\phi}{\dd x_{1}} + \frac{\dd x_{2}}{\dd x_{1}'}\, \frac{\dd\phi}{\dd x_{2}} + \frac{\dd x_{3}}{\dd x_{1}'}\, \frac{\dd\phi}{\dd x_{3}} + \frac{\dd x_{4}}{\dd x_{1}'}\, \frac{\dd\phi}{\dd x_{4}}; \text{ etc.} \] which may be written shortly \[ \frac{\dd\phi}{\dd x_{\mu}'} = \sum_{\alpha=1}^{4} \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, \frac{\dd\phi}{\dd x_{\alpha}}. \] Any set of four quantities transformed according to this law is called a \emph{covariant vector}. Thus if $A_{\mu}$~is a covariant vector, its transformation law is \index{Covariant vector}% \[ A_{\mu}' = \sum_{\alpha=1}^{4} \frac{\dd x_{\alpha}}{\dd x_{\mu}'}\, A_{\alpha}. \Tag{(19.2)} \] \PageSep{44} We have thus two varieties of vectors which we distinguish by the upper or lower position of the suffix. The first illustration of a contravariant vector, \index{Contravariant vectors}% \index{Vector!mathematical notion of}% $dx_{\mu}$, forms rather an awkward exception to the rule that a lower suffix indicates covariance and an upper suffix contravariance. There is no other exception likely to mislead the reader, so that it is not difficult to keep in mind this peculiarity of~$dx_{\mu}$; but we shall sometimes find it convenient to indicate its contravariance explicitly by writing \[ dx_{\mu} \equiv (dx)^{\mu}. \Tag{(19.3)} \]