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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Do ...
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
A common example of a vector-valued function is one that depends on a single
real parameter ''t'', often representing
time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, t ...
, producing a
vector v(''t'') as the result. In terms of the standard
unit vectors i, j, k of
Cartesian , these specific types of vector-valued functions are given by expressions such as
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
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of the domains of the functions ''f'', ''g'', and ''h''. It can also be referred to in a different notation:
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
near ''t'' = 19.5 (between 6π and 6.5π; i.e., somewhat more than 3 rotations). The
helix
A helix () is a shape like a corkscrew or spiral staircase. It is a type of smooth space curve with tangent lines at a constant angle to a fixed axis. Helices are important in biology, as the DNA molecule is formed as two intertwined hel ...
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
or
Linear case
In the
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
case the function can be expressed in terms of
matrices:
:
where ''y'' is an ''n'' × 1 output vector, ''x'' is a ''k'' × 1 vector of inputs, and ''A'' is an ''n'' × ''k'' matrix of
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. Closely related is the affine case (linear up to a
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
) where the function takes the form
:
where in addition ''b'' is an ''n'' × 1 vector of parameters.
The linear case arises often, for example in
multiple regression, where for instance the ''n'' × 1 vector
of predicted values of a
dependent variable
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or dema ...
is expressed linearly in terms of a ''k'' × 1 vector
(''k'' < ''n'') of estimated values of model parameters:
:
in which ''X'' (playing the role of ''A'' in the previous generic form) is an ''n'' × ''k'' matrix of fixed (empirically based) numbers.
Parametric representation of a surface
A
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
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric ...
s, in which two parameters ''s'' and ''t'' determine the three
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
of any point on the surface:
:
Here ''F'' is a vector-valued function. For a surface embedded in ''n''-dimensional space, one similarly has the representation
:
Derivative of a three-dimensional vector function
Many vector-valued functions, like
scalar-valued function
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
s, can be
differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if
is a vector-valued function, then
The vector derivative admits the following physical interpretation: if r(''t'') represents the
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
of a particle, then the derivative is the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
of the particle
Likewise, the derivative of the velocity is the
acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by ...
Partial derivative
The
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
of a vector function a with respect to a scalar variable ''q'' is defined as
where ''a''
''i'' is the ''scalar component'' of a in the direction of e
''i''. It is also called the
direction cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the contributions of each component of the basis to a ...
of a and e
''i'' or their
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...
. The vectors e
1, e
2, e
3 form an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
fixed in the
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
ordinary time derivative of a with respect to ''t'',
Total derivative
If the vector a is a function of a number ''n'' of scalar variables ''q''
''r'' (''r'' = 1, ..., ''n''), and each ''q''
''r'' 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
In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with r ...
, as
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''
''r'' .
Reference frames
Whereas for scalar-valued functions there is only a single possible
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
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 vectors e
1, e
2, e
3 are constant, that is, fixed in the reference frame in which the derivative of a is being taken, and therefore the e
1, e
2, e
3 each has a derivative of identically zero. This often holds true for problems dealing with
vector fields in a fixed coordinate system, or for simple problems in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which rel ...
. 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
1, e
2, e
3 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
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
1, e
2, e
3 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
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of the two reference frames
cross multiplied with the vector a itself.
Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is
where
N''ω''
E is the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
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
rocket
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entire ...
, in the
inertial reference frame using measurements of the rocket's velocity relative to the ground. The velocity
Nv
R in inertial reference frame N of a rocket R located at position r
R can be found using the formula
where
N''ω''
E is the
angular velocity
In physics, angular velocity or rotational velocity ( or ), also known as angular frequency vector,(UP1) is a pseudovector representation of how fast the angular position or orientation of an object changes with time (i.e. how quickly an objec ...
of the Earth relative to the inertial frame N. Since velocity is the derivative of position,
Nv
R and
Ev
R are the derivatives of r
R in reference frames N and E, respectively. By substitution,
where
Ev
R 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
derivative of a product of scalar functions.
[In fact, these relations are derived applying the product rule componentwise.] Specifically, in the case of
scalar multiplication of a vector, if ''p'' is a scalar variable function of ''q'',
In the case of
dot multiplication, for two vectors a and b that are both functions of ''q'',
Similarly, the derivative of the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
of two vector functions is
Derivative of an ''n''-dimensional vector function
A function ''f'' of a real number ''t'' with values in the space
can be written as
. Its derivative equals
:
.
If ''f'' is a function of several variables, say of
, then the partial derivatives of the components of ''f'' form a
matrix called the ''
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables ...
of f''.
Infinite-dimensional vector functions
If the values of a function ''f'' lie in an
infinite-dimensional vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
''X'', such as a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
,
then ''f'' may be called an ''infinite-dimensional vector function''.
Functions with values in a Hilbert space
If the
argument
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialect ...
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:
:
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.,
or even
, 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
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
) can be computed componentwise: if
:
(i.e.,
, where
is an
orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For ex ...
of the space ''X'' ), and
exists, then
:
.
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
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...
of the Hilbert space.
Other infinite-dimensional vector spaces
Most of the above hold for other
topological vector spaces ''X'' too. However, not as many classical results hold in the
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
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.
See also
*
Coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensio ...
*
Vector field
*
Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
*
Multivalued function
In mathematics, a multivalued function, also called multifunction, many-valued function, set-valued function, is similar to a function, but may associate several values to each input. More precisely, a multivalued function from a domain to ...
*
Parametric surface A parametric surface is a surface in the Euclidean space \R^3 which is defined by a parametric equation with two parameters Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that o ...
*
Position vector
In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
*
Parametrization
Notes
References
*
*
External links
Vector-valued functions and their properties (from Lake Tahoe Community College)*{{MathWorld, urlname=VectorFunction, title=Vector Function
Everything2 article*
ttp://www.khanacademy.org/video/position-vector-valued-functions?playlist=Calculus "Position Vector Valued Functions"Khan Academy
Khan Academy is an American non-profit educational organization created in 2008 by Sal Khan. Its goal is creating a set of online tools that help educate students. The organization produces short lessons in the form of videos. Its website also i ...
module
Linear algebra
Vector calculus
Vectors (mathematics and physics)
Types of functions