Suppose that vector a varies with time, so that a = a(t). The time derivative of the vector is defined

 (1.1)

When written out in component form this becomes

 (1.2)

Note that da/dt is often written in shorthand as   Suppose that a is, in fact, the product of a scalar ϕ(t) and another vector b(t). What now is the time derivative of a? We have

 (1.3)

which implies that

 (1.4)

It is easily demonstrated that

 (1.5)

Likewise,

 (1.6)

It can be seen that the laws of vector differentiation are analogous to those in conventional calculus.