vector-valued function
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
(1) |
Limit, derivative and integral of the function (1) are defined componentwise through the equations
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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
(2) |
is a (continuous) curve in . It follows from the above definition of the derivative that is the limit of the expression
(3) |
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 .
Title | vector-valued function |
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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 |