Scalar-Input Vector-Output Function

A Scalar-Input Vector-Output Function is a Real-Input Function that is a Vector-Output Function.

• AKA: Vector-Valued Real Function.
• Context:
  • domain: Real Numbers.
  • range: a Vector Space.
• Example(s):
  • [math]\displaystyle{ f(1.5) \rightarrow (3.1, 12.5) }[/math].
  • …
• Counter-Example(s):
  • a Vector-Valued Tuple Function.
  • a Vector-Input Function.
  • a Tuple-Output Function.
  • a Category-Input Vector-Output Function.
• See: Parametric Equation.

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References

2009

• (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

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