View source for Vector-valued function

{{Short description|Function valued in a vector space; typically a real or complex one}}
A '''vector-valued function''', also referred to as a '''vector function''', is a [[Function (mathematics)|mathematical function]] of one or more [[Variable (mathematics)|variables]] whose [[Range of a function|range]] is a set of multidimensional [[Vector (mathematics and physics)|vectors]] or infinite-dimensional vectors. The input of a vector-valued function could be a scalar or a vector (that is, the [[Physics:Dimension|dimension]] of the [[Domain of a function|domain]] could be 1 or greater than 1); the dimension of the function's domain has no relation to the dimension of its range.

==Example: Helix==
[[Image:Vector-valued function-2.png|300px|thumb|right|A graph of the vector-valued function {{math|1='''r'''(''z'') = &lang;2&thinsp;cos&thinsp;''z'',&thinsp;4&thinsp;sin&thinsp;''z'', ''z''&rang;}} indicating a range of solutions and the vector when evaluated near {{math|1=''z'' = 19.5}}]]

A common example of a vector-valued function is one that depends on a single [[Real number|real]] parameter ''t'', often representing [[Time|time]], producing a [[Euclidean vector|vector]] '''v'''(''t'') as the result.  In terms of the standard [[Unit vector|unit vector]]s '''i''', '''j''', '''k''' of Cartesian {{nowrap|3-space}}, these specific types of vector-valued functions are given by expressions such as
<math display="block">\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}</math>
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 (set theory)|intersection]] of the domains of the functions ''f'', ''g'', and ''h''. It can also be referred to in a different notation:
<math display="block">\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle</math>
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 <math>\langle 2\cos t,\, 4\sin t,\, t\rangle</math> near ''t''&nbsp;=&thinsp;19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The [[Helix|helix]] is the path traced by the tip of the vector as ''t'' increases from zero through 8''π''.

In 2D, We can analogously speak about vector-valued functions as
<math display="block">\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}</math> or
<math display="block">\mathbf{r}(t)=\langle f(t), g(t)\rangle</math>

==Linear case==

In the [[Linear map|linear]] case the function can be expressed in terms of [[Matrix (mathematics)|matrices]]:
:<math>y = Ax,</math>
where ''y'' is an ''n''&nbsp;×&thinsp;1 output vector, ''x'' is a ''k''&nbsp;×&thinsp;1 vector of inputs, and ''A'' is an ''n''&nbsp;×&nbsp;''k'' matrix of [[Parameter|parameter]]s. Closely related is the affine case (linear up to a [[Translation (geometry)|translation]]) where the function takes the form
:<math>y = Ax+b,</math>
where in addition ''b'' is an ''n''&nbsp;×&thinsp;1 vector of parameters.

The linear case arises often, for example in multiple regression{{clarify|reason = Is the affine case also common in multiple regression? If so please edit this paragraph accordingly.|date=December 2021}}, where for instance the ''n''&nbsp;×&thinsp;1 vector <math>\hat{y}</math> of predicted values of a dependent variable is expressed linearly in terms of a ''k''&nbsp;×&thinsp;1 vector <math>\hat{\beta}</math> (''k'' < ''n'') of estimated values of model parameters:

:<math>\hat{y} = X\hat{\beta},</math>

in which ''X'' (playing the role of ''A'' in the previous generic form) is an ''n''&nbsp;×&nbsp;''k'' matrix of fixed (empirically based) numbers.

==Parametric representation of a surface==

A [[Surface (mathematics)|surface]] is a 2-dimensional set of points embedded in (most commonly) 3-dimensional space. One way to represent a surface is with [[Parametric equation|parametric equation]]s, in which two parameters ''s'' and ''t'' determine the three Cartesian coordinates of any point on the surface:
:<math>(x, y, z) = (f(s,t), g(s,t), h(s,t)) \equiv F(s,t).</math>
Here ''F'' is a vector-valued function. For a surface embedded in ''n''-dimensional space, one similarly has the representation
:<math>(x_1, x_2, ..., x_n) = (f_1(s,t), f_2(s,t), ..., f_n(s,t)) \equiv F(s,t).</math>

==Derivative of a three-dimensional vector function==
Many vector-valued functions, like scalar-valued functions, can be [[Derivative|differentiated]] by simply differentiating the components in the Cartesian coordinate system.  Thus, if
<math display="block">\mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k}</math>
is a vector-valued function, then
<math display="block">\frac{d\mathbf{r}}{dt} = f'(t) \mathbf{i} + g'(t) \mathbf{j} + h'(t) \mathbf{k}.</math>
The vector derivative admits the following physical interpretation: if '''r'''(''t'') represents the [[Position (vector)|position]] of a particle, then the derivative is the [[Physics:Velocity|velocity]] of the particle
<math display="block">\mathbf{v}(t) = \frac{d\mathbf{r}}{dt}.</math>
Likewise, the derivative of the velocity is the [[Physics:Acceleration|acceleration]]
<math display="block">\frac{d \mathbf v}{dt} = \mathbf{a}(t).</math>

===Partial derivative===
The [[Partial derivative|partial derivative]] of a vector function '''a''' with respect to a scalar variable ''q'' is defined as<ref name="dynon19">{{harvnb|Kane|Levinson|1996|pp=29–37}}</ref> 
<math display="block">\frac{\partial\mathbf{a}}{\partial q} = \sum_{i=1}^{n}\frac{\partial a_i}{\partial q} \mathbf{e}_i</math>
where ''a''<sub>''i''</sub> is the ''scalar component'' of '''a''' in the direction of '''e'''<sub>''i''</sub>.  It is also called the [[Direction cosine#Cartesian coordinates|direction cosine]] of '''a''' and '''e'''<sub>''i''</sub> or their [[Dot product|dot product]].  The vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> form an [[Orthonormal basis|orthonormal basis]] fixed in the [[Astronomy:Frame of reference|reference frame]] in which the derivative is being taken.

===Ordinary derivative===
If '''a''' is regarded as a vector function of a single scalar variable, such as time ''t'', then the equation above reduces to the first [[Derivative|ordinary time derivative]] of '''a''' with respect to ''t'',<ref name="dynon19"/>
<math display="block">\frac{d\mathbf{a}}{dt} = \sum_{i=1}^{n}\frac{da_i}{dt} \mathbf{e}_i.</math>

===Total derivative===
If the vector '''a''' is a function of a number ''n'' of scalar variables ''q''<sub>''r''</sub> (''r''&nbsp;=&thinsp;1, ..., ''n''), and each ''q''<sub>''r''</sub> is only a function of time ''t'', then the ordinary derivative of '''a''' with respect to ''t'' can be expressed, in a form known as the [[Total derivative|total derivative]], as<ref name="dynon19"/>
<math display="block">\frac{d\mathbf a}{dt} = \sum_{r=1}^{n} \frac{\partial \mathbf a}{\partial q_r} \frac{dq_r}{dt} + \frac{\partial \mathbf a}{\partial t}.</math>

Some authors prefer to use capital ''D'' to indicate the total derivative operator, as in ''D''/''Dt''.  The total derivative differs from the partial time derivative in that the total derivative accounts for changes in '''a''' due to the time variance of the variables ''q''<sub>''r''&hairsp;</sub>.

===Reference frames===
Whereas for scalar-valued functions there is only a single possible [[Astronomy:Frame of reference|reference frame]], to take the derivative of a vector-valued function requires the choice of a reference frame (at least when a fixed Cartesian coordinate system is not implied as such).  Once a reference frame has been chosen, the derivative of a vector-valued function can be computed using techniques similar to those for computing derivatives of scalar-valued functions. A different choice of reference frame will, in general, produce a different derivative function. The derivative functions in different reference frames have a specific [[Vector-valued function#Derivative of a vector function with nonfixed bases|kinematical relationship]].

===Derivative of a vector function with nonfixed bases===
The above formulas for the derivative of a vector function rely on the assumption that the [[Basis (linear algebra)|basis]] vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are constant, that is, fixed in the reference frame in which the derivative of '''a''' is being taken, and therefore the '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> each has a derivative of identically zero.  This often holds true for problems dealing with [[Vector field|vector field]]s in a fixed coordinate system, or for simple problems in [[Physics:Physics|physics]].  However, many complex problems involve the derivative of a vector function in multiple moving reference frames, which means that the basis vectors will not necessarily be constant.  In such a case where the basis vectors '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are fixed in reference frame E, but not in reference frame N, the more general formula for the ordinary time derivative of a vector in reference frame N is<ref name="dynon19"/>
<math display="block">\frac{{}^\mathrm{N}d\mathbf{a}}{dt} = \sum_{i=1}^{3} \frac{da_i}{dt} \mathbf{e}_i + \sum_{i=1}^{3} a_i \frac{{}^\mathrm{N}d\mathbf{e}_i}{dt}</math>
where the superscript N to the left of the derivative operator indicates the reference frame in which the derivative is taken.  As shown previously, the first term on the right hand side is equal to the derivative of '''a''' in the reference frame where '''e'''<sub>1</sub>, '''e'''<sub>2</sub>, '''e'''<sub>3</sub> are constant, reference frame E.  It also can be shown that the second term on the right hand side is equal to the relative [[Physics:Angular velocity|angular velocity]] of the two reference frames cross multiplied with the vector '''a''' itself.<ref name="dynon19"/>  Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is<ref name="dynon19"/>
<math display="block">\frac{{}^\mathrm Nd\mathbf a}{dt} =  \frac{{}^\mathrm Ed\mathbf a}{dt} + {}^\mathrm N \mathbf \omega^\mathrm E \times \mathbf a</math>
where <sup>N</sup>'''''ω'''''<sup>E</sup> is the [[Physics:Angular velocity|angular velocity]] of the reference frame E relative to the reference frame N.

One common example where this formula is used is to find the velocity of a space-borne object, such as a [[Engineering:Rocket|rocket]], in the [[Physics:Inertial reference frame|inertial reference frame]] using measurements of the rocket's velocity relative to the ground.  The velocity <sup>N</sup>'''v'''<sup>R</sup> in inertial reference frame N of a rocket R located at position '''r'''<sup>R</sup> can be found using the formula
<math display="block"> \frac{{}^\mathrm Nd}{dt}(\mathbf r^\mathrm R) = \frac{{}^\mathrm Ed}{dt}(\mathbf r^\mathrm R) + {}^\mathrm N \mathbf \omega^\mathrm E \times \mathbf r^\mathrm R.</math>
where <sup>N</sup>'''''ω'''''<sup>E</sup> is the [[Physics:Angular velocity|angular velocity]] of the Earth relative to the inertial frame N.  Since velocity is the derivative of position, <sup>N</sup>'''v'''<sup>R</sup> and <sup>E</sup>'''v'''<sup>R</sup> are the derivatives of '''r'''<sup>R</sup> in reference frames N and E, respectively.  By substitution, 
<math display="block">{}^\mathrm N \mathbf v^\mathrm R =  {}^\mathrm E \mathbf v^\mathrm R + {}^\mathrm N \mathbf \omega^\mathrm E \times \mathbf r^\mathrm R</math>
where <sup>E</sup>'''v'''<sup>R</sup> is the velocity vector of the rocket as measured from a reference frame E that is fixed to the Earth.

===Derivative and vector multiplication===
The derivative of a product of vector functions behaves similarly to the [[Product rule|derivative of a product]] of scalar functions.<ref>In fact, these relations are derived applying the [[Product rule|product rule]] componentwise.</ref>  Specifically, in the case of scalar multiplication of a vector, if ''p'' is a scalar variable function of ''q'',<ref name="dynon19"/>
<math display="block">\frac{\partial}{\partial q}(p\mathbf a) = \frac{\partial p}{\partial q}\mathbf a + p\frac{\partial \mathbf a}{\partial q}.</math>

In the case of dot multiplication, for two vectors '''a''' and '''b''' that are both functions of ''q'',<ref name="dynon19"/>
<math display="block">\frac{\partial}{\partial q}(\mathbf a \cdot \mathbf b) = \frac{\partial \mathbf a }{\partial q} \cdot \mathbf b + \mathbf a \cdot \frac{\partial \mathbf b}{\partial q}.</math>

Similarly, the derivative of the cross product of two vector functions is<ref name="dynon19"/>
<math display="block">\frac{\partial}{\partial q}(\mathbf a \times \mathbf b) = \frac{\partial \mathbf a }{\partial q} \times \mathbf b + \mathbf a \times \frac{\partial \mathbf b}{\partial q}.</math>

==Derivative of an ''n''-dimensional vector function==
A function ''f'' of a real number ''t'' with values in the space <math>\R^n</math> can be written as <math>f(t)=(f_1(t),f_2(t),\ldots,f_n(t))</math>. Its derivative equals
:<math>f'(t)=(f_1'(t),f_2'(t),\ldots,f_n'(t))</math>.
If ''f'' is a function of several variables, say of <math>t\in\R^m</math>, then the partial derivatives of the components of ''f'' form a <math>n\times m</math> matrix called the ''Jacobian matrix of f''.

==Infinite-dimensional vector functions==

If the values of a function ''f'' lie in an [[Dimension (vector space)|infinite-dimensional]] [[Vector space|vector space]] ''X'', such as a [[Hilbert space]],
then ''f'' may be called an ''infinite-dimensional vector function''.

===Functions with values in a Hilbert space===

If the [[Argument of a function|argument]] of ''f'' is a real number and ''X'' is a Hilbert space, then the derivative of ''f'' at a point ''t'' can be defined as in the finite-dimensional case:
:<math>f'(t)=\lim_{h\rightarrow0}\frac{f(t+h)-f(t)}{h}.</math>
Most results of the finite-dimensional case also hold in the infinite-dimensional case too, mutatis mutandis. Differentiation can also be defined to functions of several variables (e.g., <math>t\in\R^n</math> or even <math>t\in Y</math>, where ''Y'' is an infinite-dimensional vector space).

N.B. If ''X'' is a Hilbert space, then one can easily show that any derivative (and any other [[Limit (mathematics)|limit]]) can be computed componentwise: if
:<math>f = (f_1,f_2,f_3,\ldots)</math>
(i.e., <math>f = f_1 e_1+f_2 e_2+f_3 e_3+\cdots</math>, where <math>e_1,e_2,e_3,\ldots</math> is an [[Orthonormal basis|orthonormal basis]] of the space ''X''&hairsp;), and <math>f'(t)</math> exists, then
:<math>f'(t) = (f_1'(t),f_2'(t),f_3'(t),\ldots)</math>.
However, the existence of a componentwise derivative does not guarantee the existence of a derivative, as componentwise convergence in a Hilbert space does not guarantee convergence with respect to the actual [[Topological space|topology]] of the Hilbert space.

===Other infinite-dimensional vector spaces===
Most of the above hold for other [[Topological vector space|topological vector space]]s ''X'' too. However, not as many classical results hold in the [[Banach space]] setting, e.g., an absolutely continuous function with values in a suitable Banach space need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.

==Vector field==
{{excerpt|Vector field}}

==See also==
*[[Coordinate vector]]
*[[Curve]]
*[[Multivalued function]]
*[[Parametric surface]]
*Position vector
*[[Parametrization (geometry)|Parametrization]]

==Notes==
{{Reflist}}

==References==
* {{citation|last1=Kane|first1=Thomas R.|last2=Levinson|first2=David A.|title=Dynamics Online|publisher=OnLine Dynamics, Inc.|location=Sunnyvale, California|year=1996|pages=29–37|chapter=1–9 Differentiation of Vector Functions}}
* {{citation |first=Chuang-Gan |last=Hu |first2=Chung-Chun |last2=Yang |title=Vector-Valued Functions and their Applications |publisher=Springer Science & Business Media |year=2013 |isbn=978-94-015-8030-4 |url=https://books.google.com/books?id=yuP6CAAAQBAJ }}

==External links==
*[http://ltcconline.net/greenl/courses/202/vectorFunctions/vectorFunctions.htm Vector-valued functions and their properties (from Lake Tahoe Community College)]
*{{MathWorld| urlname=VectorFunction| title=Vector Function}}
*[http://www.everything2.com/index.pl?node_id=1525585 Everything2 article]
*[http://math.etsu.edu/MultiCalc/Chap1/Chap1-6/part1.htm 3 Dimensional vector-valued functions (from East Tennessee State University)]
*[http://www.khanacademy.org/video/position-vector-valued-functions?playlist=Calculus "Position Vector Valued Functions"] Khan Academy module


[[Category:Linear algebra]]
[[Category:Vector calculus]]
[[Category:Vectors (mathematics and physics)]]
[[Category:Types of functions]]

{{Sourceattribution|Vector-valued function}}