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.
Example: Helix
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
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 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
Linear case
In 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 parameters. Closely related is the affine case (linear up to a translation) 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[clarification needed], where for instance the n × 1 vector of predicted values of a dependent variable 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 equations, in which two parameters s and t determine the three Cartesian coordinates 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 functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if
Partial derivative
The partial derivative of a vector function a with respect to a scalar variable q is defined as[1]
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,[1]
Total derivative
If the vector a is a function of a number n of scalar variables qr (r = 1, ..., n), and each qr 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[1]
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 qr .
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 e1, e2, e3 are constant, that is, fixed in the reference frame in which the derivative of a is being taken, and therefore the e1, e2, e3 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. 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 e1, e2, e3 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[1]
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 NvR in inertial reference frame N of a rocket R located at position rR can be found using the formula
Derivative and vector multiplication
The derivative of a product of vector functions behaves similarly to the derivative of a product of scalar functions.[2] Specifically, in the case of scalar multiplication of a vector, if p is a scalar variable function of q,[1]
In the case of dot multiplication, for two vectors a and b that are both functions of q,[1]
Similarly, the derivative of the cross product of two vector functions is[1]
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 of f.
Infinite-dimensional vector functions
If the values of a function f lie in an infinite-dimensional 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 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) can be computed componentwise: if
(i.e., , where is an orthonormal basis 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 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 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
- Vector field
- Curve
- Multivalued function
- Parametric surface
- Position vector
- Parametrization
Notes
References
- Kane, Thomas R.; Levinson, David A. (1996), "1–9 Differentiation of Vector Functions", Dynamics Online, Sunnyvale, California: OnLine Dynamics, Inc., pp. 29–37
- Hu, Chuang-Gan; Yang, Chung-Chun (2013), Vector-Valued Functions and their Applications, Springer Science & Business Media, ISBN 978-94-015-8030-4