A nearly similar result is obtained for tensors. The integral \[ \int T^{\mu\nu} \sqrt{-g}\, d\tau \] over an absolutely defined region is not a tensor; because, although it is the sum of a number of tensors, these tensors are not located at the same point and cannot be combined (\SecRef{33}). But in the limit as the region is made infinitely small its transformation law approaches more and more nearly that of a single tensor. Thus $T^{\mu\nu} \sqrt{-g}$~is a \emph{tensor-density}, representing the amount \index{Tensor-density}% per unit mesh of a tensor in the infinitesimal region round the point. It is usual to represent the tensor-density corresponding to any tensor by \index{German letters, denoting tensor-densities}% the corresponding German letter; thus \[ \mf{T}^{\mu\nu} \equiv T^{\mu\nu} \sqrt{-g};\quad \mf{T} \equiv T \sqrt{-g}. \Tag{(49.5)} \] By \Eq{(48.1)} \[ \mf{E}^{\alpha\beta\gamma\delta} \equiv E^{\alpha\beta\gamma\delta} \sqrt{-g} = E \sqrt{-g} · \epsilon_{\alpha\beta\gamma\delta}, \] and since $E \sqrt{-g}$ is an invariant it follows that $\epsilon_{\alpha\beta\gamma\delta}$~is a tensor-density. Physical quantities are of two main kinds, e.g.\ \index{Intensity and quantity}% \index{Quantity and intensity}% \begin{alignat*}{2} &\text{Field of acceleration} &&= \text{\emph{intensity} of some condition at a point,} \\ &\text{Momentum} &&= \text{\emph{quantity} of something in a volume.} \end{alignat*} The latter kind are naturally expressed as ``so much per unit mesh.'' Hence \emph{intensity} is naturally described by a tensor, and \emph{quantity} by a tensor-density. We shall find $\sqrt{-g}$~continually appearing in our formulae; that is an indication that the physical quantities concerned are strictly tensor-densities rather than tensors. In the general theory tensor-densities are at least as important as tensors. We can only speak of the amount of momentum in a large volume when a definite system of coordinates has been fixed. The total momentum is the sum of the momenta in different elements of volume; and for each element there will be different coefficients of transformation, when a change of coordinates is made. The only case in which we can state the amount of something in a large region without fixing a special system of coordinates is when we are dealing with an invariant; e.g.\ the amount of ``Action'' in a large region is independent of the coordinates. In short, tensor-analysis (except in the degenerate case of invariants) deals with things located at a point and not spread over a large region; that is why we usually have to use densities instead of quantities.