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TYPES OF FUNCTIONS AND FIELDS IN WHICH THEY OCCUR

I Types of functions

Type 1. Assignment of a real number to a real number.

Example. The function y = f(x) assigns a real number y to a real number x. The domain is some specified set of real numbers and the range is some set of real numbers.

Type 2. Assignment of a real number to a point in n-dimensional space.

Example 1. The function z = f(x, y) assigns a real number z to a point (x, y) in two dimensional space. The domain is some specified point-set in two dimensional space and the range is a set of real numbers.

Example 2. The function u = φ(x, y, z) assigns a real number u to a point (x, y, z) in three dimensional space. The domain is some specified point-set in three dimensional space and the range is some set of real numbers.

Example 3. The function u = φ(x1, x2, ... , xn) assigns a real number u to a point (x1, x2, ... , xn) in n-dimensional space. The domain is some specified point-set in n-dimensional space and the range is some set of real numbers.

Note. Type 1 above can be viewed as a special case of Type 2 since the function y = f(x) can be viewed as assigning a real number y to a point in one dimensional space.

Type 3. Assignment of a point in n-dimensional space to a real number.

Example 1. The system of equations

u1 = f1(t)

u2 = f2(t)

u3 = f3(t)

assigns a point (u1, u2, u3) in three dimensional space to a real number t. The domain is some specified set of real numbers and the range is a point-set in three dimensional space. Geometrically, the above function represents a curve in three dimensional space.

Example 2. The system of equations

u1 = f1(t)

u2 = f2(t)

.....

un = fn(t)

assigns a point (u1, u2, ... , un) in n-dimensional space to a real number t. The domain is some specified set of real numbers and the range is a point-set in n-dimensional space. Geometrically, the above function represents a curve in n-dimensional space.

Type 4. Assignment of a point in m-dimensional space to a point in n-dimensional space.

Example 1. The system of equations

u1 = f1(x1, x2, ... , xn)

u2 = f2(x1, x2, ... , xn)

u3 = f3(x1, x2, ... , xn)

assigns a point (u1, u2, u3) in three dimensional space to a point (x1, x2, ... , xn) in n-dimensional space. The domain is some specified point-set in n-dimensional space and the range is some point-set in three dimensional space.

Example 2. The system of equations

u1 = f1(x1, x2, ... , xn)

u2 = f2(x1, x2, ... , xn)

.....

um = fm(x1, x2, ... , xn)

assigns a point (u1, u2, ... , um) in m-dimensional space to a point (x1, x2, ... , xn) in n-dimensional space. The domain is some specified point-set in n-dimensional space and the range is some point-set in m-dimensional space.

This can also be viewed as a mapping. One can say that the system of equations maps the point (x1, x2, ... , xn) in n-dimensional space into the point (u1, u2, ... , um) in m-dimensional space.

Type 5. Assignment of a complex number to a complex number.

Note. This type of function is really a special case of Type 4 above where a point (u, v) in two dimensional space is assigned to a point (x, y) in two dimensional space.

Example 1. The function u = ax2 + bx + c , where a, b, c and x are complex numbers, assigns the complex number u to the complex number x.

Example 2. The system of equations

u = f1(x, y)

v = f2(x, y)

where x, y, u and v are real numbers assigns the pair (u, v) to the pair (x, y). This can be viewed as the assignment of a complex number (u, v) to a complex number (x, y). The domain is some specified set of complex numbers in the x-y plane and the range is a set of complex numbers in the u-v plane. This can also be viewed as a mapping -- a mapping of the complex number (x, y) into the complex number (u, v).

Type 6. Assignment of a complex number to a point in n-dimensional complex space.

Example. The function

u = f(x1, x2, ... , xn)

where x1, x2, ... , xn and u are complex numbers, assigns a complex number u to the point (x1, x2, ... , xn) in n-dimensional complex space. The domain is some specified point-set in n-dimensional complex space and the range is some set of complex numbers.

Note. This type of function is really a special case of Type 4 above. It amounts to the assignment of a point in two dimensional space to a point in a space of dimension 2n.

Type 7. Assignment of a point in n-dimensional complex space to a complex number.

Example. The system of equations

u1 = f1(t)

u2 = f2(t)

.....

un = fn(t) ,

where t is a complex number, assigns a point (u1, u2, ... , un) in n-dimensional space to a complex number t. The domain is some specified set of complex numbers and the range is some point-set in m-dimensional complex space.

Type 8. Assignment of a point in m-dimensional complex space to a point in n-dimensional complex space.

Example. The system of equations

u1 = f1(x1, x2, ... , xn)

u2 = f2(x1, x2, ... , xn)

.....

um = fm(x1, x2, ... , xn) ,

where x1, x2, ... , xn, u1, u2, ... , um are complex numbers, assigns a point (u1, u2, ... , um) in m-dimensional complex space to a point (x1, x2, ... , xn) in n-dimensional complex space. The domain is some specified point-set in n-dimensional complex space and the range is some point-set in m-dimensional complex space.

This can also be viewed as a mapping. One can say that the system of equations maps the point (x1, x2, ... , xn) in n-dimensional complex space into the point (u1, u2, ... , um) in m-dimensional complex space.

II Functions encountered in various fields

Vector Analysis

1] Type 2. Scalar point function, φ(x, y, z).

2] Type 3. Space curves.

u1 = f1(t)

u2 = f2(t)

u3 = f3(t)

3] Type 4. Vector point function.

u1 = f1(x1, x2, x3, t)

u2 = f2(x1, x2, x3, t)

u3 = f3(x1, x2, x3, t)

Theory of functions of a complex variable

1] Type 5.

Differential Geometry

1] Type 3. Curves in three dimensional space.

x = f1(t)

y = f2(t)

z = f3(t)

2] Type 4. Surfaces in three dimensional space.

x = f1(u, v)

y = f2(u, v)

z = f3(u, v)

Topology

1] Type 4. A mapping

u = f1(x, y, z)

v = f2(x, y, z)

w = f3(x, y, z)

of a figure in three dimensional space into a distorted version of the same figure without tearing.