Higher Concepts of Mathematics
Many practical problems can be solved using arithmetic and algebra; however,
many other practical problems involve quantities that cannot be adequately
described using numbers which have fixed values.
STATE the graphical definition of a derivative.
STATE the graphical definition of an integral.
Arithmetic involves numbers that have fixed values. Algebra involves both literal and arithmetic
numbers. Although the literal numbers in algebraic problems can change value from one
calculation to the next, they also have fixed values in a given calculation. When a weight is
dropped and allowed to fall freely, its velocity changes continually. The electric current in an
alternating current circuit changes continually. Both of these quantities have a different value
at successive instants of time. Physical systems that involve quantities that change continually
are called dynamic systems. The solution of problems involving dynamic systems often involves
mathematical techniques different from those described in arithmetic and algebra. Calculus
involves all the same mathematical techniques involved in arithmetic and algebra, such as
addition, subtraction, multiplication, division, equations, and functions, but it also involves several
other techniques. These techniques are not difficult to understand because they can be developed
using familiar physical systems, but they do involve new ideas and terminology.
There are many dynamic systems encountered in nuclear facility work. The decay of radioactive
materials, the startup of a reactor, and a power change on a turbine generator all involve
quantities which change continually. An analysis of these dynamic systems involves calculus.
Although the operation of a nuclear facility does not require a detailed understanding of calculus,
it is most helpful to understand certain of the basic ideas and terminology involved. These ideas
and terminology are encountered frequently, and a brief introduction to the basic ideas and
terminology of the mathematics of dynamic systems is discussed in this chapter.
Differentials and Derivatives
One of the most commonly encountered applications of the mathematics of dynamic systems
involves the relationship between position and time for a moving object. Figure 2 represents an
object moving in a straight line from position P1 to position P2. The distance to P1 from a fixed
reference point, point 0, along the line of travel is represented by S1; the distance to P2 from
point 0 by S2.