stylistic:style_transfer
A tensor equation represents many ordinary (real-valued) equations

The components of tensors (like A, B etc.) are just real numbers. Since the indices take various integer values to select specific components of the tensors, a single tensor equation represents many ordinary equations. If a tensor equality has n free indices, and if the dimensionality of the underlying vector space is m, the equality represents m equations: each index takes on every value of a specific set of values.

For instance, if

is in four dimensions (that is, each index runs from 0 to 3 or from 1 to 4), then because there are three free indices (α, β, δ), there are 4 = 64 equations. Three of these are:

This illustrates the compactness and efficiency of using index notation: many equations which all share a similar structure can be collected into one simple tensor equation.