Vector-valued function explained

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, often representing time, producing a vector as the result. In terms of the standard unit vectors,, of Cartesian , these specific types of vector-valued functions are given by expressions such as$$\backslash mathbf(t)\; =\; f(t)\backslash mathbf\; +\; g(t)\backslash mathbf\; +\; h(t)\backslash mathbf$$where, and are the coordinate functions of the parameter, and the domain of this vector-valued function is the intersection of the domains of the functions,, and . It can also be referred to in a different notation:$$\backslash mathbf(t)\; =\; \backslash langle\; f(t),\; g(t),\; h(t)\backslash rangle$$The vector 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

\langle 2 \cos t, 4 \sin t, t\rangle near (between and ; i.e., somewhat more than 3 rotations). The helix is the path traced by the tip of the vector as increases from zero through .

In 2D, We can analogously speak about vector-valued functions as$$\backslash mathbf(t)\; =\; f(t)\backslash mathbf\; +\; g(t)\backslash mathbf$$ or$$\backslash mathbf(t)\; =\; \backslash langle\; f(t),\; g(t)\backslash rangle$$

Linear case

In the linear case the function can be expressed in terms of matrices:$$\backslash mathbf\; =\; A\backslash mathbf,$$where is an output vector, is a vector of inputs, and is an matrix of parameters. Closely related is the affine case (linear up to a translation) where the function takes the form$$\backslash mathbf\; =\; A\backslash mathbf\; +\; \backslash mathbf,$$where in addition is an vector of parameters.

The linear case arises often, for example in multiple regression, where for instance the vector

\hat{y} of predicted values of a dependent variable is expressed linearly in terms of a vector \hat{\boldsymbol\beta} of estimated values of model parameters:$$\backslash hat\; =\; X\backslash hat,$$in which (playing the role of in the previous generic form) is an 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 and determine the three Cartesian coordinates of any point on the surface:$$(x,\; y,\; z)\; =\; (f(s,t),\; g(s,t),\; h(s,t))\; \backslash equiv\; \backslash mathbf(s,t).$$Here is a vector-valued function. For a surface embedded in -dimensional space, one similarly has the representation$$(x\_1,\; x\_2,\; \backslash dots,\; x\_n)\; =\; (f\_1(s,t),\; f\_2(s,t),\; \backslash dots,\; f\_n(s,t))\; \backslash equiv\; \backslash mathbf(s,t).$$

Derivative of a three-dimensional vector function

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$$\backslash mathbf(t)\; =\; f(t)\; \backslash mathbf\; +\; g(t)\; \backslash mathbf\; +\; h(t)\; \backslash mathbf$$is a vector-valued function, then$$\backslash frac\; =\; f\text{'}(t)\; \backslash mathbf\; +\; g\text{'}(t)\; \backslash mathbf\; +\; h\text{'}(t)\; \backslash mathbf.$$The vector derivative admits the following physical interpretation: if represents the position of a particle, then the derivative is the velocity of the particle$$\backslash mathbf(t)\; =\; \backslash frac.$$Likewise, the derivative of the velocity is the acceleration$$\backslash frac\; =\; \backslash mathbf(t).$$

Partial derivative

The partial derivative of a vector function with respect to a scalar variable is defined as^[1] $$\backslash frac\; =\; \backslash sum\_^\backslash frac\; \backslash mathbf\_i$$where is the scalar component of in the direction of . It is also called the direction cosine of and or their dot product. The vectors,, form an orthonormal basis fixed in the reference frame in which the derivative is being taken.

Ordinary derivative

If is regarded as a vector function of a single scalar variable, such as time, then the equation above reduces to the first ordinary time derivative of a with respect to,^[1] $$\backslash frac\; =\; \backslash sum\_^\backslash frac\; \backslash mathbf\_i.$$

Total derivative

If the vector is a function of a number of scalar variables, and each is only a function of time, then the ordinary derivative of with respect to can be expressed, in a form known as the total derivative, as^[1] $$\backslash frac\; =\; \backslash sum\_^\; \backslash frac\; \backslash frac\; +\; \backslash frac.$$

Some authors prefer to use capital to indicate the total derivative operator, as in . The total derivative differs from the partial time derivative in that the total derivative accounts for changes in due to the time variance of the variables .

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] $$\backslash frac\; =\; \backslash sum\_^\; \backslash frac\; \backslash mathbf\_i\; +\; \backslash sum\_^\; a\_i\; \backslash frac$$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 in the reference frame where,, 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 multiplied with the vector a itself.^[1] Thus, after substitution, the formula relating the derivative of a vector function in two reference frames is^[1] $$\backslash frac\; =\; \backslash frac\; +\; ^\backslash mathrm\; N\; \backslash mathbf\; \backslash omega^\backslash mathrm\; E\; \backslash times\; \backslash mathbf\; a$$where is the 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 rocket, in the inertial reference frame using measurements of the rocket's velocity relative to the ground. The velocity in inertial reference frame N of a rocket R located at position can be found using the formula$$\backslash frac(\backslash mathbf\; r^\backslash mathrm\; R)\; =\; \backslash frac(\backslash mathbf\; r^\backslash mathrm\; R)\; +\; ^\backslash mathrm\; N\; \backslash mathbf\; \backslash omega^\backslash mathrm\; E\; \backslash times\; \backslash mathbf\; r^\backslash mathrm\; R.$$where is the angular velocity of the Earth relative to the inertial frame N. Since velocity is the derivative of position, and are the derivatives of in reference frames N and E, respectively. By substitution, $$^\backslash mathrm\; N\; \backslash mathbf\; v^\backslash mathrm\; R\; =\; ^\backslash mathrm\; E\; \backslash mathbf\; v^\backslash mathrm\; R\; +\; ^\backslash mathrm\; N\; \backslash mathbf\; \backslash omega^\backslash mathrm\; E\; \backslash times\; \backslash mathbf\; r^\backslash mathrm\; R$$where 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. Specifically, in the case of scalar multiplication of a vector, if is a scalar variable function of,^[1] $$\backslash frac(p\backslash mathbf\; a)\; =\; \backslash frac\backslash mathbf\; a\; +\; p\backslash frac.$$

In the case of dot multiplication, for two vectors and that are both functions of,^[1] $$\backslash frac(\backslash mathbf\; a\; \backslash cdot\; \backslash mathbf\; b)\; =\; \backslash frac\; \backslash cdot\; \backslash mathbf\; b\; +\; \backslash mathbf\; a\; \backslash cdot\; \backslash frac.$$

Similarly, the derivative of the cross product of two vector functions is^[1] $$\backslash frac(\backslash mathbf\; a\; \backslash times\; \backslash mathbf\; b)\; =\; \backslash frac\; \backslash times\; \backslash mathbf\; b\; +\; \backslash mathbf\; a\; \backslash times\; \backslash frac.$$

Derivative of an n-dimensional vector function

A function of a real number with values in the space

\R ^{n} can be written as f (t) = (f _{1(t),f} _{2(t),\ldots,f} _{n(t))} . Its derivative equals$$\backslash mathbf\text{'}(t)\; =\; (f\_1\text{'}(t),\; f\_2\text{'}(t),\; \backslash ldots,\; f\_n\text{'}(t)).$$If is a function of several variables, say of then the partial derivatives of the components of form a n x m matrix called the Jacobian matrix of .

Infinite-dimensional vector functions

See main article: Infinite-dimensional-vector function.

If the values of a function lie in an infinite-dimensional vector space, such as a Hilbert space, then may be called an infinite-dimensional vector function.

Functions with values in a Hilbert space

If the argument of is a real number and is a Hilbert space, then the derivative of at a point can be defined as in the finite-dimensional case:$$\backslash mathbf\text{'}(t)\; =\; \backslash lim\_\; \backslash frac.$$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.,

t \in \R ^{n} or even t \in Y , where is an infinite-dimensional vector space).

N.B. If is a Hilbert space, then one can easily show that any derivative (and any other limit) can be computed componentwise: if$$\backslash mathbf\; =\; (f\_1,\; f\_2,f\_3,\backslash ldots)$$(i.e., where

e _{1,} e _{2,} e _{3,} \ldots is an orthonormal basis of the space  ), and f'(t) exists, then$$\backslash mathbf\text{'}(t)\; =\; (f\_1\text{'}(t),\; f\_2\text{'}(t),\; f\_3\text{'}(t),\; \backslash ldots).$$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 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
• Curve
• Multivalued function
• Parametric surface
• Position vector
• Parametrization

References

• Book: Hu, Chuang-Gan . Chung-Chun . Yang . Vector-Valued Functions and their Applications . Springer Science & Business Media . 2013 . 978-94-015-8030-4 .

External links

• Vector-valued functions and their properties (from Lake Tahoe Community College)
  • Everything2 article
• 3 Dimensional vector-valued functions (from East Tennessee State University)
• "Position Vector Valued Functions" Khan Academy module

Notes and References

1. Book: Kane . Thomas R. . Levinson . David A. . Dynamics: Theory and Applications . McGraw-Hill . Sunnyvale, California . 1996 . 29–37 . 1–9 Differentiation of Vector Functions.