Vector-Valued Function

A Vector-Valued Function is a Function whose Function Range is a Vector Set.

• AKA: Vector-Output Function.
• Context:
  • Function Range is a Vector Set (possibly from a Vector Space).
  • It can be:
    • a Vector-Valued Real Function.
    • ...
• Example(s):
  • f(1.5) => (3.1,2.3).
  • f(1.1,Yellow,3.9) => (π,2.3).
• Counter-Example(s):
  • f(1.1,Yellow,3.9) => π, a Real-Valued Function.
  • f(1.1,2.3,3.9) => Red, a Category-Valued Function.
• See: Vector-Input Function, Tuple-Output Function, Parametric Equation.

References

• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Vector-valued_function
  • A vector-valued function is a mathematical function that maps real numbers to vectors. Vector-valued functions can be defined as:
    • \mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}} or
    • \mathbf{r}(t)=f(t)\mathbf{{\hat{i}}}+g(t)\mathbf{{\hat{j}}}+h(t)\mathbf{{\hat{k}}}
    • where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and \mathbf{{\hat{i}}}, \mathbf{{\hat{j}}}, and \mathbf{{\hat{k}}} are unit vectors. r(t) is a vector which has its tail at the origin and its head at the coordinates evaluated by the function.
  • Properties: The domain of a vector-valued function is the intersection of the domain of the functions f, g, and h.
• http://ltcconline.net/greenl/courses/202/vectorFunctions/vecfun.htm
  • A vector valued function is a function where the domain is a subset of the real numbers and the range is a vector.
  • In two dimensions: r(t) = x(t)i + y(t)j
  • In three dimensions: r(t) = x(t)i + y(t)j + z(t)k
  • You will notice the strong resemblance to parametric equations. In fact there is an equivalence between vector valued functions and parametric equations.
• http://ltcconline.net/greenl/courses/202/vectorIntegration/vectorFields.htm#fields
  • We have now seen many types of functions. They are characterized by the domain and the range.
  • Below is a list of some of the functions that we have encountered so far.
    

    Domain	Range	Name
    R	R	One variable Function
    R	R2	Parametric Equations
    R2	R	Function of 2 Variables
    R	Vectors	Vector Valued Function