Vector-valued function

A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the dimension of the domain could be 1 or greater than 1); the dimension of the function’s domain has no relation to the dimension of its range.

1 Example: Helix

Further information: Parametric curve

A common example of a vector-valued function is one that depends on a single real parameter t, often representing time, producing a vector v(t) as the result. In terms of the standard unit vectors i, j, k of Cartesian 3-space, these specific types of vector-valued functions are given by expressions such as

{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} +h(t)\mathbf {k} }
where f(t), g(t) and h(t) are the coordinate functions of the parameter t, and the domain of this vector-valued function is the intersection of the domains of the functions f, g, and h. It can also be referred to in a different notation:
{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }{\displaystyle \mathbf {r} (t)=\langle f(t),g(t),h(t)\rangle }
The vector r(t) has its tail at the origin and its head at the coordinates evaluated by the function.
The vector shown in the graph to the right is the evaluation of the function {\displaystyle \langle 2\cos t,,4\sin t,,t\rangle }{\displaystyle \langle 2\cos t,,4\sin t,,t\rangle } near t = 19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The helix is the path traced by the tip of the vector as t increases from zero through 8π.

In 2D, We can analogously speak about vector-valued functions as

{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} }{\displaystyle \mathbf {r} (t)=f(t)\mathbf {i} +g(t)\mathbf {j} }
or
{\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }{\displaystyle \mathbf {r} (t)=\langle f(t),g(t)\rangle }

A graph of the vector-valued function r(z) = ⟨2 cos z, 4 sin z, z⟩ indicating a range of solutions and the vector when evaluated near z = 19.5

2 Linear case

3 Parametric representation of a surface

4 Derivative of a three-dimensional vector function

4.1 Partial derivative

4.2 Ordinary derivative

4.3 Total derivative

4.4 Reference frames

4.5 Derivative of a vector function with nonfixed bases

4.6 Derivative and vector multiplication

5 Derivative of an n-dimensional vector function

6 Infinite-dimensional vector functions

6.1 Functions with values in a Hilbert space

6.2 Other infinite-dimensional vector spaces

7 See also