Vector function

Derivative of a three-dimensional vector function

Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if $${\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }$$ is a vector-valued function, then $${\displaystyle {\frac {d\mathbf {r} }{dt}}=f'(t)\mathbf {i} +g'(t)\mathbf {j} +h'(t)\mathbf {k} .}$$ The vector derivative admits the following physical interpretation: if r(t) represents the position of a particle, then the derivative is the velocity of the particle $${\displaystyle \mathbf {v} (t)={\frac {d\mathbf {r} }{dt}}.}$$ Likewise, the derivative of the velocity is the acceleration$${\displaystyle {\frac {d\mathbf {v} }{dt}}=\mathbf {a} (t).}$$