Let us now pass to one of the most important characteristics of tensors and vectors. The fact that the components of all contravariant vectors are submitted to the same rules of change when we pass from one co-ordinate system to another, proves that if two contravariant vectors situated at the same point of space are equal, that is, if the components of the one are equal to the components of the other in our co-ordinate system, this equality will inevitably endure in any other co-ordinate system. In other words, the equality of two vectors at a point of space constitutes an equality which a change in our co-ordinate system can never destroy; the equality is thus an absolute. Similar conclusions apply to the equality of two tensors at the same point of space.
It must be noted, however, that this equality endures only because the components of either vector or of either tensor on the two sides of the equation are submitted to the same rules of transformation when our co-ordinate system is changed. This implies that we are dealing with vectors of the same nature (covariant or contravariant), or with tensors of the same nature and order. If these conditions are satisfied, we see that equations between vectors or tensors, often called vector equations and tensor equations, exhibit the remarkable property of remaining unaffected by a change of mesh-system.
In a certain sense, then, vector equations and tensor equations constitute invariant or absolute equations. This last appellation would, in fact, be perfectly justified were we to argue in terms of the absolute vectors and tensors which exist independently of our mesh-system. But in practice it is very difficult to dispense with a mesh-system. As a result, what we come into contact with—what we measure in reality—is not the vector or the tensor itself, which is absolute, but its components in the mesh-system that we have selected. These components are relative, since they change in value when we change our mesh-system. True, these changes being the same on either side of the equation, the equality endures regardless of our choice of mesh-system. But, on the other hand, a definite numerical change has taken place in the value of either side of our equation. The form of the equation as a whole has remained unaltered, but the contents of the two sides has changed.
Let us explain this point more fully. Consider two vectors which are situated at the same point and which are equal to each other. In three-dimensional space, this equality is expressed by the respective equality of the three components of the first vector with the three components of the second vector in our co-ordinate system; hence we have three equations, such as
If, now, we change our mesh-system, the components of either vector change in exactly the same way, so that our equations subsist. In the new mesh-system, however,
becomes
and