For these reasons it is customary to speak of the covariance rather than of the invariance of vector or tensor equations. The word “covariance” expresses the fact that both sides have varied in exactly the same way. The appellation “invariance” would tend to make us believe that neither side had varied at all, which would, of course, be incorrect. In short, we must remember that it is the relationship of equality between the two sides, and not the individual terms themselves, that remain unaffected by a change of mesh-system.[89]
Now, there exists still another type of vector and of tensor equation. If, for instance, a vector or a tensor vanishes, this means that all the components vanish individually. A simple glance at the rules of transformation shows that in this case all the components will continue to vanish in any co-ordinate system, so that the equating of a tensor, a vector or an invariant to zero at a point of space will endure in all co-ordinate systems if it endures in any particular system.
Summarising, we find that there exist two types of covariant equations:
The first type is given by the equality of two vectors (the vectors being both covariant or both contravariant) or, more generally, by the equality of two tensors, these tensors being of the same order and same nature (i.e., covariant, mixed or contravariant). A limiting case would be given by the equality of two invariants; we should then have an invariant equation.
The second type is given by the vanishing of a vector or a tensor at a point in space. A limiting case would be given by the vanishing of an invariant.
All these deductions obtained mathematically are in a certain sense self-evident physically when we recall that a tensor, a vector and an invariant represent magnitudes to which an objective existence may be conceded. It is obvious, indeed, that if two such magnitudes are equal to each other at a point in space, a mere change of our co-ordinate system can never disturb this intrinsic equality. In the same way, if a concrete entity, such as a tensor, has vanished at a point in space, a mere change of our co-ordinate system is incapable of bringing it into existence. Once again, we realise the strict dependence which exists between physical representations of this sort and the fact that invariants, vectors and tensors must be recognised to have an objective existence transcending the particular point of view of our co-ordinate system.
If these mathematical considerations have been grasped, we may pass to physical applications of these purely mathematical discoveries. We will restrict ourselves for the present to the three-dimensional Euclidean space of classical science; hence by a change of co-ordinate system we again mean a mere change in shape, position and orientation of our mesh-system. We do not mean the passage from one system to another in relative motion thereto.
In classical science we often come upon invariants and vectors. Examples of invariants are given by the mass and the density of mass of a body. Vectors are illustrated by a velocity, an acceleration, a force, a displacement; and we may say that in a general way a vector can be represented by an arrow. The simplest physical illustration of a tensor at each point of space is afforded by the state of compression and tension at each point of an elastic medium which has been distorted.
It is impossible to express this state of tension and compression at a point by means of three components in our three-dimensional space. Owing to lateral wrenching effects at each point, nine components are necessary; and we realise that we can no longer appeal to a vector. We are now in the presence of a tensor. The physical origin of the appellation “tensor,” coined by Willard Gibbs, thus becomes apparent. In the present case the tensor is a second-order tensor, and, furthermore, it is what is known as a symmetrical tensor. The characteristics of a symmetrical tensor are defined by the identity of certain of its components two by two; this reduces the number of independent components of a second-order tensor in three-dimensional space to six in place of nine.
It can be proved that the