The result that the covariant curl is the same as the ordinary curl does not apply to contravariant vectors or to tensors of higher rank: \[ {K^{\mu}}_{\nu} - {K^{\nu}}_{\mu} \neq \frac{\dd K^{\mu}}{\dd x_{\nu}} - \frac{\dd K^{\nu}}{\dd x_{\mu}}. \] In tensor notation the famous theorem of Stokes becomes \index{Stokes's theorem}% \[ \int K_{\mu}\, dx_{\mu} = -\frac{1}{2} \iint \left(\frac{\dd K_{\mu}}{\dd x_{\nu}} - \frac{\dd K_{\nu}}{\dd x_{\mu}}\right) dS^{\mu\nu}, \Tag{(32.3)} \] the double integral being taken over any surface bounded by the path of the single integral. The factor~$\frac{1}{2}$ is needed because each surface-element occurs twice, e.g.\ as $dS^{12}$ and $-dS^{21}$. The theorem can be proved as follows--- Since both sides of the equation are invariants it is sufficient to prove the equation for any one system of coordinates. Choose coordinates so that the \PageSep{68} surface is on one of the fundamental partitions $x_{2} = \text{const.}$, $x_{4} = \text{const.}$, and so that the contour consists of four parts given successively by $x_{1} = \alpha$, $x_{2} = \beta$, $x_{1} = \gamma$, $x_{2} = \delta$; the rest of the mesh-system may be filled up arbitrarily. For an elementary mesh the containing vectors are $(dx_{1}, 0, 0, 0)$ and $(0, dx_{2} , 0, 0)$, so that by~\Eq{(32.1)} \[ dS^{12} = dx_{1}\, dx_{2} = -dS^{21}. \] Hence the right-hand side of~\Eq{(32.3)} becomes \begin{multline*} -\int_{\alpha}^{\gamma} \int_{\beta}^{\delta} \left(\frac{\dd K_{1}}{\dd x_{2}} - \frac{\dd K_{2}}{\dd x_{1}}\right) dx_{1}\, dx_{2} \\ = -\int_{\alpha}^{\gamma} \bigl\{[K_{1}]^{\delta} - [K_{1}]^{\beta}\bigr\}\, dx_{1} + \int_{\beta}^{\delta} \bigl\{[K_{2}]^{\gamma} - [K_{2}]^{\alpha}\bigr\}\, dx_{2}, \end{multline*} which consists of four terms giving $\int K_{\mu}\, dx_{\mu}$ for the four parts of the contour. This proof affords a good illustration of the methods of the tensor calculus. The relation to be established is between two quantities which (by examination of their covariant dimensions) are seen to be invariants, viz.\ $K_{\mu} (dx)^{\mu}$ and \index{Covariant derivative of vector!significance of}% \index{Derivative!significance of}% $(K_{\mu\nu} - K_{\nu\mu})\, dS^{\mu\nu}$, the latter having been simplified by~\Eq{(32.2)}. Accordingly it is a relation which does not depend on any particular choice of coordinates, although in~\Eq{(32.3)} it is expressed as it would appear when referred to a coordinate-system. In proving the relation of the two invariants once for all, we naturally choose for the occasion coordinates which simplify the analysis; and the work is greatly shortened by drawing our curved meshes so that four partition-lines make up the contour. \Section{33.}{Significance of covariant differentiation}