Scalar-Input Vector-Output Function
(Redirected from Vector-Valued Real Function)
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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):
- See: Parametric Equation.
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.
- A vector-valued function is a mathematical function that maps real numbers to vectors. Vector-valued functions can be defined as:
- 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.
R | R | One variable Function |
R | R2 | Parametric Equations |
R2 | R | Function of 2 Variables |
R | Vectors | Vector Valued Function |