Double\footnote {The doubling of the natural expression is avenged by the appearance of the factor~$\frac{1}{2}$ in most formulae containing~$dS^{\mu\nu}$.} the antisymmetrical part of the product~$dx_{\mu}\, \delta x_{\nu}$ is called the \emph{surface-element} contained by the two displacements, and is denoted by~$dS^{\mu\nu}$. \index{Surface-element}% We have accordingly \begin{align*} dS^{\mu\nu} &= dx_{\mu}\, \delta x_{\nu} - dx_{\nu}\, \delta x_{\mu} \Tag{(32.1)} \\ &= \left\lvert\begin{array}{@{}cc@{}} dx_{\mu} & dx_{\nu} \\ \delta x_{\mu} & \delta x_{\nu} \\ \end{array}\right\rvert. \end{align*} \PageSep{67} In rectangular coordinates this determinant represents the area of the projection on the $\mu\nu$~plane of the parallelogram contained by the two displacements; thus the components of the tensor are the projections of the parallelogram on the six coordinate planes. In the tensor~$dS^{\mu\nu}$ these are repeated twice, once with positive and once with negative sign (corresponding perhaps to the two sides of the surface). The four components $dS^{11}$, $dS^{22}$, etc.\ vanish, as must happen in every antisymmetrical tensor. The appropriateness \index{Antisymmetrical tensors}% of the name ``surface-element'' is evident in rectangular coordinates; the geometrical meaning becomes more obscure in other systems. The surface-element is always a tensor of the second rank whatever the number of dimensions of space; but in \emph{three} dimensions there is an alternative representation of a surface area by a simple \emph{vector} at right angles to the surface and of length proportional to the area; indeed it is customary in three dimensions to represent any antisymmetrical tensor by an adjoint vector. Happily in four dimensions it is not possible to introduce this source of confusion. The invariant \[ \tfrac{1}{2} A_{\mu\nu}\, dS^{\mu\nu} \] is called the \emph{flux} of the tensor~$A_{\mu\nu}$ through the surface-element. The flux \index{Flux}% involves only the antisymmetrical part of~$A_{\mu\nu}$, since the inner product of a symmetrical and an antisymmetrical tensor evidently vanishes. Some of the chief antisymmetrical tensors arise from the operation of \emph{curling}. If $K_{\mu\nu}$~is the covariant derivative of~$K_{\mu}$, we find from~\Eq{(29.3)} that \[ K_{\mu\nu} - K_{\nu\mu} = \frac{\dd K_{\mu}}{\dd x_{\nu}} - \frac{\dd K_{\nu}}{\dd x_{\mu}} \Tag{(32.2)} \] since the $3$-index symbols cancel out. Since the left-hand side is a tensor, the right-hand side is also a tensor. The right-hand side will be recognised as the ``curl'' of elementary vector theory, except that we have apparently reversed \index{Curl}% the sign. Strictly speaking, however, we should note that the curl in the elementary three-dimensional theory is a vector, whereas our curl is a tensor; and comparison of the sign attributed is impossible.