vector-valued function

Let n be a positive integer greater than 1.  A functionMathworldPlanetmath F from a subset T of to the Cartesian product n is called a vector-valued function of one real variable.  Such a function to any real number t of T a coordinate vector


Hence one may say that the vector-valued function F is composed of n real functionstfi(t),  the values of which at t are the components of F(t).  Therefore the function F itself may be written in the component form

F=(f1,,fn). (1)

Example.  The ellipsePlanetmathPlanetmath


is the value set of a vector-valued function  2  (t is the eccentric anomaly).

Limit, derivative and integralDlmfPlanetmath of the function (1) are defined componentwise through the equations

  • limtt0F(t):=(limtt0f1(t),,limtt0fn(t))

  • F(t):=(f1(t),,fn(t))

  • abF(t)𝑑t:=(abf1(t)𝑑t,,abfn(t)𝑑t)

The function F is said to be continuousMathworldPlanetmath, differentiableMathworldPlanetmathPlanetmath or integrable on an intervalMathworldPlanetmathPlanetmath[a,b]  if every component of F has such a property.

Example.  If F is continuous on  [a,b],  the set

γ:={F(t)t[a,b]} (2)

is a (continuous) curve in n.  It follows from the above definition of the derivative F(t) that F(t) is the limit of the expression

1h[F(t+h)-F(t)] (3)

as  h0.  Geometrically, the vector (3) is parallelMathworldPlanetmathPlanetmathPlanetmath to the line segmentMathworldPlanetmath connecting (the end points of the position vectors of) the points F(t+h) and F(t).  If F is differentiable in t, the direction of this line segment then tends infinitely the direction of the tangent lineMathworldPlanetmath of γ in the point F(t).  Accordingly, the direction of the tangent line is determined by the derivative vector F(t).

Title vector-valued function
Canonical name VectorvaluedFunction
Date of creation 2013-03-22 19:02:19
Last modified on 2013-03-22 19:02:19
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Definition
Classification msc 26A36
Classification msc 26A42
Classification msc 26A24
Related topic Component
Related topic DifferenceOfVectors
Defines integrable