Website owner: James Miller
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.