Vector-valued function

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From Wikipedia, the free encyclopedia

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[edit]

Further information: Parametric curve

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

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

$${\displaystyle \mathbf{�}(�)=�(�)\mathbf{�}+�(�)\mathbf{�}+ℎ(�)\mathbf{�}}$$

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:

$${\displaystyle \mathbf{�}(�)=⟨�(�),�(�),ℎ(�)⟩}$$

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 ${\displaystyle ⟨2\cos �,4\sin �,�⟩}$ 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

$${\displaystyle \mathbf{�}(�)=�(�)\mathbf{�}+�(�)\mathbf{�}}$$

or

$${\displaystyle \mathbf{�}(�)=⟨�(�),�(�)⟩}$$

Linear case[edit]

In the linear case the function can be expressed in terms of matrices:

${\displaystyle �=��,}$

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

${\displaystyle �=��+�,}$

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 ${\displaystyle \widehat{�}}$ of predicted values of a dependent variable is expressed linearly in terms of a k × 1 vector ${\displaystyle \widehat{�}}$ (k < n) of estimated values of model parameters:

${\displaystyle \widehat{�}=�\widehat{�},}$

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[edit]

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:

${\displaystyle (�,�,�)=(�(�,�),�(�,�),ℎ(�,�))\equiv �(�,�).}$

Here F is a vector-valued function. For a surface embedded in n-dimensional space, one similarly has the representation

${\displaystyle ({�}_1,{�}_2,...,{�}_{�})=({�}_1(�,�),{�}_2(�,�),...,{�}_{�}(�,�))\equiv �(�,�).}$

Derivative of a three-dimensional vector function[edit]

See also: Gradient

Many vector-valued functions, like scalar-valued functions, can be differentiated by simply differentiating the components in the Cartesian coordinate system. Thus, if

$${\displaystyle \mathbf{�}(�)=�(�)\mathbf{�}+�(�)\mathbf{�}+ℎ(�)\mathbf{�}}$$

is a vector-valued function, then

$${\displaystyle \frac{�\mathbf{�}}{��}={�}^{\prime }(�)\mathbf{�}+{�}^{\prime }(�)\mathbf{�}+{ℎ}^{\prime }(�)\mathbf{�}.}$$

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

$${\displaystyle \mathbf{�}(�)=\frac{�\mathbf{�}}{��}.}$$

Likewise, the derivative of the velocity is the acceleration

$${\displaystyle \frac{�\mathbf{�}}{��}=\mathbf{�}(�).}$$

Partial derivative[edit]

The partial derivative of a vector function a with respect to a scalar variable q is defined as^[1]

$${\displaystyle \frac{\mathrm{\partial }\mathbf{�}}{\mathrm{\partial }�}=\sum _{�=1}^{�}\frac{\mathrm{\partial }{�}_{�}}{\mathrm{\partial }�}{\mathbf{�}}_{�}}$$

where a_i is the scalar component of a in the direction of e_i. It is also called the direction cosine of a and e_i or their dot product. The vectors e₁, e₂, e₃ form an orthonormal basis fixed in the reference frame in which the derivative is being taken.

Ordinary derivative[edit]

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]

$${\displaystyle \frac{�\mathbf{�}}{��}=\sum _{�=1}^{�}\frac{�{�}_{�}}{��}{\mathbf{�}}_{�}.}$$

Total derivative[edit]

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, as^[1]

$${\displaystyle \frac{�\mathbf{�}}{��}=\sum _{�=1}^{�}\frac{\mathrm{\partial }\mathbf{�}}{\mathrm{\partial }{�}_{�}}\frac{�{�}_{�}}{��}+\frac{\mathrm{\partial }\mathbf{�}}{\mathrm{\partial }�}.}$$

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[edit]

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.