Editing Four-vector (section)

== Four-vector calculus ==

===Derivatives and differentials===

In special relativity (but not general relativity), the [[derivative]] of a four-vector with respect to a scalar ''λ'' (invariant) is itself a four-vector. It is also useful to take the [[differential of a function|differential]] of the four-vector, ''d'''''A''' and divide it by the differential of the scalar, ''dλ'':

<math display="block">\underset{\text{differential}}{d\mathbf{A}} = \underset{\text{derivative}}{\frac{d\mathbf{A}}{d\lambda}} \underset{\text{differential}}{d\lambda} </math>

where the contravariant components are:

<math display="block"> d\mathbf{A} = \left(dA^0, dA^1, dA^2, dA^3\right) </math>

while the covariant components are:

<math display="block"> d\mathbf{A} = \left(dA_0, dA_1, dA_2, dA_3\right) </math>

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