Let be a positive integer greater than 1. A function from a subset of to the Cartesian product is called a vector-valued function of one real variable. Such a function to any real number of a coordinate vector
Hence one may say that the vector-valued function is composed of real functions , the values of which at are the components of . Therefore the function itself may be written in the component form
Example. The ellipse
is the value set of a vector-valued function ( is the eccentric anomaly).
Limit, derivative and integral of the function (1) are defined componentwise through the equations
The function is said to be continuous, differentiable or integrable on an interval if every component of has such a property.
Example. If is continuous on , the set
is a (continuous) curve in . It follows from the above definition of the derivative that is the limit of the expression
as . Geometrically, the vector (3) is parallel to the line segment connecting (the end points of the position vectors of) the points and . If is differentiable in , the direction of this line segment then tends infinitely the direction of the tangent line of in the point . Accordingly, the direction of the tangent line is determined by the derivative vector .
|Date of creation||2013-03-22 19:02:19|
|Last modified on||2013-03-22 19:02:19|
|Last modified by||pahio (2872)|