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| A '''vector-valued function''', also referred to as a '''vector function''', is a [[function (mathematics)|mathematical function]] of one or more variables whose [[range (mathematics)|range]] is a set of multidimensional [[Euclidean vector|vectors]] or infinite-dimensional [[infinite-dimensional-vector-valued function|vectors]]. The input of a vector-valued function could be a scalar or a vector. The dimension of the domain is not defined by the dimension of the range.
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| ==Example==
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| [[Image:Vector-valued function-2.png|300px|thumb|right|A graph of the vector-valued function '''r'''(''t'') = <{{nowrap|2 cos ''t'', 4 sin ''t'', ''t''}}> indicating a range of solutions and the vector when evaluated near {{nowrap|''t'' {{=}} 19.5}}]]
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| A common example of a vector valued function is one that depends on a single [[real number]] parameter ''t'', often representing [[time]], producing a [[Euclidean vector|vector]] '''v'''(''t'') as the result. In terms of the standard [[unit vector]]s '''i''', '''j''', '''k''' of [[Cartesian space|Cartesian 3-space]], these specific type of vector-valued functions are given by expressions such as
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| *<math>\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}</math> or
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| *<math>\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}</math>
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| where ''f''(''t''), ''g''(''t'') and ''h''(''t'') are the '''coordinate functions''' of the [[Parametric equation|parameter]] ''t''. The vector '''r'''(''t'') has its tail at the origin and its head at the coordinates evaluated by the function.
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| The vector shown in the graph to the right is the evaluation of the function near ''t''=19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The spiral is the path traced by the tip of the vector as t increases from zero through 8π.
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| Vector functions can also be referred to in a different notation:
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| *<math>\mathbf{r}(t)=\langle f(t), g(t)\rangle</math> or
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| *<math>\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle</math>
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| ==Properties==
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| The [[Domain (mathematics)|domain]] of a vector-valued function is the [[Intersection (set theory)|intersection]] of the domain of the functions ''f'', ''g'', and ''h''.
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| ==Derivative of a three-dimensional vector function==
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| {{see also|Gradient}}
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| Many vector-valued functions, like [[scalar-valued function]]s, can be [[derivative|differentiated]] by simply differentiating the components in the Cartesian coordinate system. Thus, if
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| :<math>\mathbf{r}(t) = f(t)\mathbf{i} + g(t)\mathbf{j} + h(t)\mathbf{k}</math>
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| is a vector-valued function, then
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| :<math>\frac{d\mathbf{r}(t)}{dt} = f'(t)\mathbf{i} + g'(t)\mathbf{j} + h'(t)\mathbf{k}.</math>
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| The vector derivative admits the following physical interpretation: if '''r'''(''t'') represents the position of a particle, then the derivative is the [[velocity]] of the particle
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| :<math>\mathbf{v}(t) = \frac{d\mathbf{r}(t)}{dt} .</math>
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| Likewise, the derivative of the velocity is the acceleration
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| :<math>\frac{d\bold{v}(t)}{d t}=\bold{a}(t).</math>
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| ===Partial derivative===
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| The [[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>
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| :<math>\frac{\partial\mathbf{a}}{\partial q} = \sum_{i=1}^{n}\frac{\partial a_i}{\partial q}\mathbf{e}_i</math>
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| 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]] fixed in the [[Frame of reference|reference frame]] in which the derivative is being taken.
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| ===Ordinary derivative===
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| 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 [[ordinary derivative|ordinary time derivative]] of '''a''' with respect to ''t'',<ref name="dynon19"/>
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| :<math>\frac{d\mathbf{a}}{dt} = \sum_{i=1}^{3}\frac{da_i}{dt}\mathbf{e}_i.</math>
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| ===Total derivative===
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| If the vector '''a''' is a function of a number ''n'' of scalar variables ''q''<sub>''r''</sub> (''r'' = 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]], as<ref name="dynon19"/>
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| :<math>\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>
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| 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''</sub>.
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| ===Reference frames===
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| Whereas for scalar-valued functions there is only a single possible [[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]].
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| ===Derivative of a vector function with nonfixed bases===
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| The above formulas for the derivative of a vector function rely on the assumption that the 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]]s in a fixed coordinate system, or for simple problems in 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_derivative|ordinary time derivative]] of a vector in reference frame N is<ref name="dynon19"/>
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| :<math>\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>
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| where the superscript N to the left of the derivative operator indicates the reference frame in which the derivative is taken. [[#Ordinary_derivative|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 [[angular velocity]] of the two reference frames [[#Cross_product|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"/>
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| :<math>\frac{{}^\mathrm Nd\mathbf a}{dt} = \frac{{}^\mathrm Ed\mathbf a }{dt} + {}^\mathrm N \mathbf \omega^\mathrm E \times \mathbf a</math>
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| where <sup>N</sup>'''''ω'''''<sup>E</sup> is the [[angular velocity]] of the reference frame E relative to the reference frame N.
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| One common example where this formula is used is to find the [[velocity]] of a space-borne object, such as a [[rocket]], in the [[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
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| :<math> \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>
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| where <sup>N</sup>'''''ω'''''<sup>E</sup> is the [[angular velocity]] of the Earth relative to the inertial frame N. Since [[velocity]] is the [[derivative]] of [[position (vector)|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,
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| :<math>{}^\mathrm N \mathbf v^\mathrm R = {}^\mathrm E \mathbf v^\mathrm R + {}^\mathrm N \mathbf \omega^\mathrm E \times \mathbf r^\mathrm R</math>
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| 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.
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| ===Derivative and vector multiplication===
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| The derivative of the products of vector functions behaves similarly to the [[product rule|derivative of the products]] of scalar functions.<ref>In fact, these relations are derived applying the [[product rule]] componentwise.</ref> Specifically, in the case of [[#Scalar_multiplication|scalar multiplication]] of a vector, if ''p'' is a scalar variable function of ''q'',<ref name="dynon19"/>
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| :<math>\frac{\partial}{\partial q}(p\mathbf a) = \frac{\partial p}{\partial q}\mathbf a + p\frac{\partial \mathbf a}{\partial q}.</math> | |
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| In the case of [[#Dot_product|dot multiplication]], for two vectors '''a''' and '''b''' that are both functions of ''q'',<ref name="dynon19"/>
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| :<math>\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>
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| Similarly, the derivative of the [[#Cross_product|cross product]] of two vector functions is<ref name="dynon19"/>
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| :<math>\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>
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| ==Derivative of an ''n''-dimensional vector function==
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| 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
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| :<math>f'(t)=(f_1'(t),f_2'(t),\ldots,f_n'(t))</math>.
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| 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''.
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| ==Infinite-dimensional vector functions==
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| If the values of a function ''f'' lie in an [[infinite-dimensional]] vector space ''X'', such as a [[Hilbert space]],
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| then ''f'' may be called an ''infinite-dimensional vector function''.
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| ===Functions with values in a Hilbert space===
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| If the 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:
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| :<math>f'(t)=\lim_{h\rightarrow0}\frac{f(t+h)-f(t)}{h}.</math>
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| 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).
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| N.B. If ''X'' is a Hilbert space, then one can easily show that any derivative (and any other limit) can be computed componentwise: if
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| :<math>f=(f_1,f_2,f_3,\ldots)</math>
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| (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]] of the space ''X''), and <math>f'(t)</math> exists, then
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| :<math>f'(t)=(f_1'(t),f_2'(t),f_3'(t),\ldots)</math>.
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| 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 topology of the Hilbert space.
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| ===Other infinite-dimensional vector spaces===
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| Most of the above hold for other [[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 [[Radon–Nikodym property|suitable Banach space]] need not have a derivative anywhere. Moreover, in most Banach spaces setting there are no orthonormal bases.
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| ==See also==
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| *[[Infinite-dimensional-vector function]]
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| *[[Coordinate vector]]
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| *[[Vector field]]
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| *[[Curve]]
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| *[[Parametric surface]]
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| *[[Position vector]]
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| *[[Parametrization]]
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| ==Notes==
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| <references/>
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| ==References==
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| *{{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}}
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| ==External links==
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| *[http://ltcconline.net/greenl/courses/202/vectorFunctions/vectorFunctions.htm Vector-valued functions and their properties (from Lake Tahoe Community College)]
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| *{{MathWorld| urlname=VectorFunction| title=Vector Function}}
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| *[http://www.everything2.com/index.pl?node_id=1525585 Everything2 article]
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| *[http://math.etsu.edu/MultiCalc/Chap1/Chap1-6/part1.htm 3 Dimensional vector-valued functions (from East Tennessee State University)]
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| *[http://www.khanacademy.org/video/position-vector-valued-functions?playlist=Calculus "Position Vector Valued Functions"] [[Khan Academy]] module
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| [[Category:Linear algebra]]
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| [[Category:Vector calculus]]
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| [[Category:Vectors]]
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| [[Category:Types of functions]]
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