Applied mathematics

Summary

Applied mathematics is the application of mathematical principles and theory in the real world. The practice of applied mathematics has two principal objectives—to find solutions to challenging problems by identifying the mathematical rules that describe the observed behavior or characteristic involved and to reduce real-world behaviors to a level of precise and accurate predictability. Mathematical rules and operations are devised to describe a behavior or property that may not yet have been observed, aiming to predict with certainty what the outcome of the behavior would be.

Definition and Basic Principles

Applied mathematics focuses on the development and study of mathematical and computational tools. These tools are used to solve challenging problems primarily in science and engineering applications and other fields amenable to mathematical procedures. The principal mathematical tool is calculus, often called the mathematics of change. Calculus provides a means of quantitatively understanding how variables that cannot be controlled directly behave in response to changes in variables that can be controlled directly. Thus, applied mathematics makes it possible to make predictions about the behavior of an environment and thus gain some mastery over that environment.

For example, suppose a specific characteristic of the behavior of individuals in a society is determined by the combination of several influencing forces, many of which are unknown and perhaps unknowable and, therefore, not directly controllable. The study of the occurrence of that characteristic in a population allows it to be described in mathematical terms. This, in turn, provides a valid means of predicting the future occurrence and behavior of that characteristic in other situations. Applied mathematics, therefore, uses mathematical techniques and the results of those techniques in the investigation or solving of problems that originate outside the realm of mathematics.

The applications of mathematics to real-world phenomena rely on four essential structures: data structures, algorithms, theories and models, and computers and software. Data structures are ways of organizing information or data. Algorithms are specific methods of dealing with the data. Theories and models are used in the analysis of both data and ideas and represent the rules that describe either the way the data were formed or the behavior of the data. Computers and software are the physical devices used to manipulate the data for analysis and application. Algorithms are central to the development of software, which is computer-specific, for the manipulation and analysis of data.

Background and History

Applied mathematics as a field of study is newer than engineering and building. The mathematical principles at the center of applied mathematics were developed and devised from the observation of physical constructs and behaviors and, therefore, after the development of those activities. The foundations of applied mathematics are found in the works of early Egyptian and Greek philosophers and engineers. Plane geometry is thought to have developed during the reign of Pharaoh Sesostris as a result of agricultural land measurements necessitated by the annual inundation of the Nile River. The Greek engineer Thales of Miletus is credited with some of the earliest and most profound applications of mathematical and physical principles in the construction of some of his devices, although there is no evidence that he left a written record of those principles. The primary historical figures in the development of applied mathematics are Euclid and Archimedes. The Greek method of philosophy lacked physical experimentation and the testing of hypotheses but was a pure thought process. For this reason, there is a distinction between pure and applied mathematics, although the latter depends strictly on the former.

During the Middle Ages, significant mathematical development occurred as al geber, modern-day algebra, gained scientific acceptance, but the field of mathematics showed little progress in Europe. Even during the Renaissance period, mathematics was almost exclusively for astronomers and astrologers. Despite the mathematical brilliance of his designs, it is uncertain if Leonardo da Vinci, the foremost of the Renaissance engineers and artists, was adept at mathematics. The major development of applied mathematics began with the development of calculus by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the seventeenth century. The applicability of mathematical principles in the development of scientific pursuits during the Industrial Revolution brought applied mathematics to the point where it has become essential for understanding the physical universe.

How It Works

Applied mathematics is the creation and study of mathematical and computational tools that can be broadly applied in science and engineering. Those tools are used to solve challenging problems in these and related fields. In its simplest form, applied mathematics refers to using measurement and simple calculations to describe a physical condition or behavior.

A simple example might be the layout or design of a field or other area of land. Consider a need to lay out a rectangular field having an area of 2,000 square meters (m²) with the shortest possible perimeter. The area (A) of any rectangular area is determined as the product of the length (l) and the width (w) of the area in question. The perimeter (P) of any rectangular area is determined as the sum of the lengths of all four sides, and in a rectangular area, the opposite sides are of equal length. Thus, P = 2l + 2w, and A = l × w = 2000 m². By trial and error, pairs of lengths and widths whose product is 2,000 may be tried out, and their corresponding perimeters determined. A field that is 2,000 meters long and 1 meter wide has an area of 2,000 square meters and a perimeter of 4,002 meters. Similarly, a field that is 200 meters long and 10 meters wide has the required area, and a perimeter of only 420 meters. It becomes apparent that the perimeter is minimized when the area is represented as a square, having four equal sides. Thus, the length of each side must be equal to the square root of 2,000 in magnitude. Having determined this, the same principles may be applied to the design of any rectangular area of any size.

The same essential procedures demonstrated in this simple example apply with equal validity to other physical situations and are the essence of scientific experimentation and research. The progression of the development of mathematical models and procedures in many areas of application is remarkably similar. The development of a mathematical model begins with a simple expression to which refinements are made. The results are compared with the actual behavior of the system under investigation. The changes in the difference between the real and calculated behaviors are the key to further refinements that, ideally, bring the calculated and actual behaviors into closer agreement. When mathematical expressions adequately describe the behavior of a system, those expressions can be used to describe the behaviors of other systems.

A key component to successfully applying mathematical descriptions or models is understanding variables that affect the behavior of the system being studied. In fluid dynamics, for example, the obvious variables that affect a fluid are the temperature and density of the fluid. Less obvious, perhaps, are such variables as the viscosity of the fluid, the dipolar interactions of the fluid atoms or molecules, the adhesion between the fluid and the surface of the container through which it is flowing, whether the fluid is flowing smoothly (laminar flow) or turbulently (nonlaminar flow), and several other more obscure variables. A precise mathematical description of the behavior of such a system would include corrective terms for every variable affecting the system. However, many of these corrective terms may be considered together in an approximation term and still produce an accurate mathematical description of the behavior.

An example of such an approximation may be found in the applied mathematical field of quantum mechanics, by which the behavior of electrons in molecules is modeled. The classic quantum mechanical model of the behavior of an electron bound to an atomic nucleus is the so-called particle-in-a-box model. In this model, the particle (the electron) can exist only within the confines of the box (the atomic orbital), and because the electron has the properties of an electromagnetic wave as well as those of a physical particle, there are certain restrictions placed on the behavior of the particle. For example, the value of the wave function describing the motion of the electron must be zero at the boundaries of the box. This requires that the motion of the particle can be described only by certain wave functions that, in turn, depend on the dimensions of the box. The problem can be solved mathematically with precision only for the case involving a single electron and a single nuclear proton that defines the box in which the electron is found. The calculated results agree extremely well with observed measurements of electron energy.

For systems involving more particles (more electrons and more nuclear protons and neutrons), the number of variables and other factors immediately exceeds any ability to be calculated precisely. A solution is found, however, in a method that uses an approximation of the orbital description, known as a Slater-type orbital approximation, rather than a precise mathematical description. A third-level Gaussian treatment of the Slater-type orbitals, or STO-3G analysis, yields calculated results for complex molecular structures in excellent agreement with the observed values measured in physical experiments. Although the level of mathematical technique is more complex than in the simple area example, the basic method of finding an applicable method is almost identical.

Applications and Products

Applied mathematics is essentially the application of mathematical principles and theories toward the resolution of physical problems and the description of behaviors. The range of disciplines in which applied mathematics is relevant is, therefore, very broad. The intangible nature of mathematics and mathematical theory tends to restrict active research and development to the academic environment and applied research departments of industry. In these environments, applied mathematics research is usually focused rather than general. Applied mathematics is generally divided into the major areas of computational mathematicscombinatorics and optimization, computer science, pure mathematics, and statistical and actuarial science. The breadth of the research field has grown dramatically, and the diversity of subject area applications is indicated by the applied mathematics research being conducted in atmospheric and biological systems applications, climate and weather, complexity theory, computational finance, control systems, cryptography, pattern recognition, data mining, multivariate data analysis and visualization, differential equation modeling, fluid dynamics, linear programming, medical imaging, and a host of other areas.

Computational Mathematics. Simply stated, computational mathematics is the process of modeling systems quantitatively on computers. This is often referred to in the literature as in silico, indicating that the operation or procedure being examined is carried out as a series of calculations within the silicon-based electronic circuitry of a computer chip and not in any tangible, physical manner. Research in computational mathematics is conducted in a range of subject areas.

The essence of computational mathematics is the development of algorithms and computer programs that produce accurate and reliable models or depictions of specific behaviors. In atmospheric systems, for example, one goal would be to produce mathematical programs that precisely depict the behavior of the ozone layer surrounding the planet. The objective of such a program would be to predict how the ozone layer would change as a result of alterations in atmospheric composition. It is not feasible to observe the effects directly and would ultimately be counterproductive if manifesting an atmospheric change resulted in the destruction of the ozone layer. Modeling the system in silico allows researchers to institute virtual changes and determine the effect of each change. The reliability of the calculated effect depends on how accurately the model describes the system's existing behavior.

Medical Imaging. An area in which applied mathematics has become fundamentally important is the field of medical imaging, especially as it applies to magnetic resonance imaging (MRI). The MRI technique developed from nuclear magnetic resonance (NMR) analysis is commonly used in analytical chemistry to determine molecular structures. In NMR, measurements are obtained of the absorption of specific radio frequencies by molecules held within a magnetic field. The strength of each absorption and specific patterns of absorptions are characteristic of the structure of the particular molecule and so can be used to determine unequivocally the exact molecular structure of a material.

Applied mathematics has greatly improved NMR by eliminating background noise. A typical NMR spectrum consists of an infinite series of small random signals that often hide the detailed patterns of actual absorption peaks and sometimes even the peaks themselves. The Fourier analysis methodology, in which such random signals can be treated as a combination of sine and cosine waves, also known as wavelet theory, eliminates many such signals from electromagnetic spectra. The result is a clearer and more precise record of the actual absorptions. Such basic NMR spectra are only one-dimensional, however. The second generation modification of NMR systems was developed to produce a two-dimensional representation of the NMR absorption spectrum, and from this was developed the three-dimensional NMR imaging system known as MRI. Improvements that make the Fourier analysis technique ever more effective in accord with advances in the computational abilities of computer hardware are the focus of one area of ongoing applied mathematics research.

Population Dynamics and Epidemiology. Population dynamics and epidemiology are closely related fields of study. The former studies the growth and movements of populations, and the latter studies the growth and movements of diseases and medical conditions within populations. Both rely heavily for their mathematical descriptions on many areas of applied mathematics, including statistics, fluid dynamics, complexity theory, pattern recognition, data visualization, differential equation modeling, chaos theory, risk management, numerical algorithms and techniques, and statistical learning.

In a practical model, the movements of groups of individuals within a population are described by many of the same mathematical models of fluid dynamics that apply to moving streams of particles. The flow of traffic on a multilane highway or the movement of people along a busy sidewalk, for example, can exhibit the same gross behavior as that of a fluid flowing through a system of pipes. Any population that can be described in terms of a flow of discrete particles, whether molecules of water or vehicles, can be described by the same mathematical principles, at least to the extent that the variables affecting their motion are known. Thus, the forces of friction and adhesion that affect the flow of a fluid within a tube are closely mimicked by the natural tendencies of drivers to drive at varying speeds in different lanes of a multilane highway. Window-shoppers and other slow-moving individuals tend to stay to the part of the sidewalk closest to the buildings, while those who walk faster or more purposefully tend to use the part of the sidewalk farthest away from the buildings, and this also follows the behavior of fluid flow.

The spread or movement of diseases through a population can also be described by many of the same mathematical principles that describe the movements of individuals within a population. This is especially true for diseases transmitted directly from person to person. For other disease vectors, such as animals, birds, and insects, a mathematical description must describe the movements of those particular populations, while at the same time reflecting the relationship between those populations and the human population of interest.

Statistical Analysis and Actuarial Science. Perhaps the simplest or most obvious use of applied mathematics can be found in statistical analysis. In this application, the common properties of a collection of data points, themselves measurements of some physical property, are enumerated and compared for consistency. The effectiveness of statistical methods depends on the appropriately random collection of representative data points and the appropriate definition of a property to be analyzed.

Statistical analysis is used to assess the consistency of a common property and to identify patterns of occurrence of characteristics. This forms the basis of statistical process control (SPC), which has become the primary quality control method in industry and other fields of practice. In statistical process control, random samples of an output stream are selected and compared to their design standard. Variations from the desired value are determined, and the data accumulated over time are analyzed to determine patterns of variation. In an injection-molding process, for example, a variation that occurs consistently in one location of the object being molded may indicate a modification to the overall process must be made, such as adjusting the temperature of the liquid material being injected or an alteration to the die itself to improve the plastic flow pattern.

In another context, one that is tied to epidemiology, the insurance and investment industries make very detailed use of statistical analysis in risk assessment. Massive amounts of data describing various aspects of human existence in modern society are meticulously analyzed to identify patterns of effects that may indicate a causal relationship. An obvious example is the statistical relationship pattern between a healthy lifestyle and mortality rates, in which obese people of all ages have a higher mortality rate than those who maintain a leaner body mass. Similarly, automobile insurance rates are much higher for male drivers under twenty-five than for female drivers in that age group and older drivers because statistical analysis of data from accidents demonstrates this group has the highest risk of involvement in a traffic accident. This type of data mining is a continual process as relationships are sought to describe every factor that plays a role in human society.

Careers and Course Work

The study of applied mathematics builds on a solid and in-depth comprehension of pure mathematics. The student begins by taking courses in mathematics throughout secondary school to acquire a solid foundational knowledge of basic mathematical principles before entering college or university studies. A specialization in mathematics at the college and university level is essential to most areas of study. As so many fields have been affected by applied mathematics, the methodologies taught in undergraduate courses reflect the accepted concepts of applied mathematics on which they are constructed. As the depth of the field of applied mathematics indicates, career options are, for all intents and purposes, unlimited. Every field of endeavor, from anthropology to zoology, has a component of applied mathematics, and the undergraduate must learn the mathematical methods corresponding to the chosen field.

Students who specialize in the study of applied mathematics as a career choice and proceed to postgraduate studies take courses in advanced mathematics and carry out research aimed at developing mathematics theory and advancing the application of those developments in other fields.

Particular programs of study in applied mathematics are included as part of the curriculum of other disciplines. The focus of such courses is on specific mathematical operations and techniques that are relevant to that particular field of study. All include a significant component of differential calculus, as appropriate to the dynamic nature of the subject matter.

Social Context and Future Prospects

Human behavior is well-suited and highly amenable to description and analysis through mathematical principles. Modeling of population dynamics has become increasingly important in the contexts of service and regulation. As new diseases appear, accurate models to predict how they will spread play an ever more important role in determining the response to possible outbreaks. Similarly, modeling of geological activities is increasingly valuable in determining the best ways to respond to such natural disasters as a tsunami or earthquake. Earth itself is a dynamic system that is only marginally predictable. Applied mathematics is essential to developing models and theories that provide accurate predictions of the occurrence and ramifications of seismic events. It will also be necessary to understand the effects of human activity on the atmosphere and oceans, particularly concerning climate change and the emission of greenhouse gases. Climate models that accurately and precisely predict the effects of such activities will continue to be the object of a great deal of research in applied mathematics.

Developing new materials and engineering new applications is an ongoing human endeavor. The design of tiny nanostructures that employ those materials is based on mathematical principles unique to the realm of the very small. IBM released its first circuit-based quantum computer, Osprey, in 2019, a device that operates on the subatomic scale rather than on the existing scale of computer construction. As quantum computers develop and become increasingly accessible, considerable investments in such technology will be necessary to remain competitive for businesses in many industries.

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