We call the tensor A_{μν} the "extension" of the tensor A_μ. Then we can easily show that this combination also leads to a tensor, when the vector A_μ is not representable as a gradient. In order to see this we first remark that ψ (dφ/[part]x_μ) is a co-variant four-vector when ψ and φ are scalars. This is also the case for a sum of four such terms :—
S_μ = ψ^{(1)}([part]φ^{(1)}/[part]x_μ) + . . . + ψ^{(4)}([part]φ^{(4)}/[part]x_μ)
when ψ^{(1)}, φ^{(1)} . . . ψ^{(4)} φ^{(4)} are scalars. Now it is however clear that every co-variant four-vector is representable in the form of S_μ.
If for example, A_μ is a four-vector whose components are any given functions of x_ν, we have, (with reference to the chosen co-ordinate system) only to put
ψ^{(1)} = A_1 φ^{(1)} = x_1
ψ^{(2)} = A_2 φ^{(2)} = x_2
ψ^{(3)} = A_3 φ^{(3)} = x_3
ψ^{(4)} = A_4 φ^{(4)} = x_4.
in order to arrive at the result that S_μ is equal to A_μ.
In order to prove then that A_{μν} in a tensor when on the right side of (26) we substitute any co-variant four-vector for A_μ we have only to show that this is true for the
