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of a four-vector with respect to a scalar (invariant) is itself a four-vector. It is also useful to take the differential of the four-vector, and divide it by the differential of the scalar, :

where the contravariant components are:

while the covariant components are:

In relativistic mechanics, one often takes the differential of a four-vector and divides by the differential in proper time (see below).

Fundamental four-vectors

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