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Vector function

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A vector function is any function whose output is a vector.

Parametric equations are often represented as vector functions in the following manner. This example has a single input (t) and produces three simultaneous outputs (x,y,z) all which are functions of t in themselves. The sum of the three components produces the radius vector.

\vec{F}(t)=\langle x(t),y(t),z(t)\rangle
=x(t)\mathbf{\hat{i}}+y(t)\mathbf{\hat{j}}+z(t)\mathbf{\hat{k}}

Another example would be a vector function with coordinates as inputs, in the form

\vec{F}(x,y)=p(x,y)\mathbf{\hat{i}}+q(x,y)\mathbf{\hat{j}}

This equation represents a two-dimensional vector field.

The derivative of a vector function is equal to \vec{F}'(t)=\left\langle\frac{dx}{dt},\frac{dy}{dt},\frac{dz}{dt}\right\rangle The integral of a vector function is equal to \int\vec{F}(t)dt=\left\langle\int x(t)dt,\int y(t)dt,\int z(t)dt\right\rangle

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