Engineering Mathematics
Engineering Mathematics is a vital discipline that applies mathematical principles and techniques to solve complex engineering problems, often through the development and use of mathematical models, simulations, and software tools. This field encompasses a broad range of applications, from formulating precise mathematical relationships governing physical systems to employing algorithms and computational methods. Historical figures such as Galileo and Coulomb have laid foundational theories that link mathematics with fundamental laws of physics, paving the way for subsequent advancements in engineering and technology.
Engineering mathematics facilitates the modeling of diverse physical phenomena using various types of equations, including ordinary and partial differential equations. The accuracy of these models is crucial and is assessed through rigorous computational experiments and error analysis. The discipline plays a significant role in contemporary research areas, such as cell biology, genetics, materials science, and environmental engineering, highlighting its interdisciplinary nature.
Career opportunities in engineering mathematics are expansive, covering sectors like data analysis, materials development, and ecological engineering. As technology continues to evolve, engineering mathematics remains at the forefront of innovation, addressing emerging global challenges and enhancing the performance of high-tech products through advanced computational techniques.
Engineering Mathematics
Summary
Engineering mathematics focuses on the use of mathematics as a tool within the engineering design process. Such use includes the development and application of mathematical models, simulations, computer systems, and software to solve complex engineering problems, the solution to which might be a component, system, or process.
Definition and Basic Principles
Engineering mathematics involves the development and application of mathematics—such as algorithms, models, computer systems, and software—within the engineering design process. In both coursework and professional research, various tools may be used to collect, analyze, and display data. Standard measurement tools include rules, spirit levels, micrometers, calipers, and gauges. Software tools include Maplesoft, Mathematica, MATLAB, and Excel. Engineering mathematics has roots and applications in many areas, including algorithmic science, applied mathematics, computational science, and bioinformatics.
Background and History
Since engineering is such a broad area, engineering mathematics includes a variety of applications. In general, a link between engineering and mathematics is established when mathematical descriptions of physical systems are formulated.
Links may involve precise mathematical relationships or formulas. For example, Galileo Galilei's pioneering work on the study of the motion of physical objects led to the equations of accelerated motion, ν = at and d = ½ at2, in which velocity is ν, acceleration a, time t, and distance d. His work paved the way for Newtonian physics.
Other links are established through empirical relationships that have the status of laws. French physicist Charles-Augustin de Coulomb discovered that the force between two electrical charges is proportional to the product of the charges and inversely proportional to the square of the distance between themF = kq1q2/d2 (force is F, constant of variation k, charges q1 and q2, and distance d). This came to be known as Coulomb's law, and a measure of electrical charge, the Coulomb, was also named for him. Coulomb's work was the first in a sequence of related discoveries by other notable scientists, many of whose findings led to additional laws. The list includes Danish physicist Hans Christian Ørsted; French physicists André-Marie Ampère, Jean-Baptiste Biot, and Félix Savart; British physicist and chemist Michael Faraday; and Russian physicist Heinrich Lenz.
Sometimes, mathematical expressions of principles apply almost universally. In physics, for example, the conservation laws indicate that in a closed system, certain measurable quantities remain constant: mass, momentum, energy, and mass energy. Lastly, systems of equations are required to describe physical phenomena of various levels of complexity. Examples include English astronomer and mathematician Isaac Newton's equations of motion, Scottish physicist and mathematician James Clerk Maxwell's equations for electromagnetic fields, Swiss mathematician and physicist Leonhard Euler's equations in fluid mechanics, and French engineer Claude-Louis Navier and British mathematician and physicist George Gabriel Stokes's generalization of Euler's equations, known as the Navier-Stokes equation.
Further links between engineering and mathematics are discovered through the ongoing development, extension, modification, and generalization of equations and models in broader physical systems. For example, the Euler equations used in fluid mechanics can be connected to the conservation laws of mass and momentum.
How It Works
Many problems in engineering mathematics lead to the construction of models that can be used to describe physical systems. Because of the power of technology, a model may be derived from a system of a few equations that may be linear, quadratic, exponential, or trigonometric—or a system of many equations of even greater complexity. In engineering, such equations include ordinary differential equations, differential algebraic equations, and partial differential equations.
As the system of equations is solved, the mathematical model is formulated. Models are expressed in terms of mathematical symbols and notation that represent objects or systems and the relationships between them. Computer software, such as Maplesoft, Mathematica, MATLAB, and Excel, facilitates the process.
Engineers have available many models of physical systems. The development, extension, and modification of existing models, and the development of new models, are the subject of ongoing research. In this way, engineering mathematics continues to advance.
The ultimate test of a mathematical model is whether it truly reflects the behavior of the physical system under study. Computational experiments can be run to test the model for unexpected characteristics of the system and possibly optimize its design. However, models are approximations, and the accuracy of computed results must be evaluated through some form of error analysis.
The level of complexity of the construction and use of models depends on the engineering application. Further appreciation of the utility of a model may be gained by examining the use of a new model in engineering mathematics that impacts several scientific and technological areas. A mathematical model can now be used to investigate how materials break. One led to a new law of physics that depicts fracturing before it happens, or even as it occurs. In addition to the breakage of materials such as glass and concrete used in construction, the model enables better examination of bone breakage in patients with pathologies such as osteoporosis.
Applications and Products
Cell Biology. Advances in research in cell growth and division have proved helpful in disease detection, pharmaceutical research, and cell and tissue engineering. Biologists have extensively explored cell growth and mass and the relationship between them. Using microsensors, bioengineers can now delineate colon cancer cell masses and divisions over given periods. They have found that such cells grow faster as they grow heavier. With additional cell measurements and mathematical modeling, the scientists examined other properties, such as stiffness. They also performed simulations to study the relationship between cell stiffness, contact area, and mass measurement.
Genetics. New genes involved in stem-cell development can be found, quickly and inexpensively, along the same pathways as genes already known. When searching for genes involved in a particular biological process, scientists try to find genes with a symmetrical correlation. However, many biological relationships are asymmetric and can now be found using Boolean logic in data-mining techniques. Engineering and medical researchers can then examine whether such genes become active, such as those in developing cancers. This research is expected to lead to advances in disease diagnosis and cancer therapy.
Energy. A new equation could help to further the use of organic semiconductors. The equation represents the relationship between current and voltage at the junctions of the organic semiconductors. Research in the use of organic semiconductors may lead to advances in solar cells, displays, and lighting. Engineers have been studying the building blocks of organic semiconductors since the 1860s but only began to discover innovative applications in the 1970s.
Careers and Course Work
Engineering mathematics careers are built on solid technical skills in mathematical modeling. Applications include a broad range of industries. Engineering mathematicians may help design the next Formula One racecar, develop autonomous driving software, build robotics, or create a revolutionary prediction model for infectious diseases.
Data Analyst or Data Miner. The discovery of hidden but useful information in large databases is known as data mining. In applications of data mining, career opportunities emerge in medicine, science, and engineering. Data mining involves the use of algorithms to identify and verify previously undiscovered relationships and structures from rigorous data analysis. Coursework should include a focus on higher-level mathematics in such areas as topology, combinatorics, and algebraic structures.
Materials Science. Materials science is the research, development, and manufacture of such items as metallic alloys, liquid crystals, and biological materials. There are many career opportunities in aerospace, electronics, biology, and nanotechnology. Research and development uses mathematical models and computational tools. Coursework should include a focus on applied mathematics, including differential equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, and probability.
Ecological and Environmental Engineering. Ecological and environmental engineering is a relatively new branch of engineering mathematics. Oregon State University began the first ecological engineering bachelor's degree in 2009, and the Journal of Ecological Engineering Design was established in 2021. Professionals in ecological and environmental engineering cover many areas that transcend pollution control, public health, and waste management. Professions include, for example, the design, construction, and management of an aquatic ecosystem or the research and development of appropriate sustainable technologies. Other professionals may focus on restoring ecosystems, mitigating pollution, and other complex ecological problems. Coursework includes ecology, economics, natural sciences, and engineering. Focusing on higher-level mathematics in calculus, linear algebra, differential equations, and statistics is helpful.
Meteorology and Climatology. The fields of meteorology and climatology incorporate not only atmospheric, hydrologic, and oceanographic sciences but also modeling, forecasting, geoengineering, and geophysics. In general, historical weather data and modern satellite data, radar, and monitoring equipment are combined with other measurements to develop, process, and analyze complex models using high-performance computers. Modern research areas include global warming and the impact of atmospheric radiation and industrial pollutants. Mathematics courses in meteorology and atmospheric science programs include calculus, differential equations, linear algebra, statistics, computer science, numerical analysis, and matrix algebra or computer systems.
Social Context and Future Prospects
Within engineering mathematics, an interdisciplinary specialty has emerged called computational engineering. Computational engineering employs mathematical models, numerical methods, science, engineering, and computational processes that connect various fields of engineering science. Computational engineering emerged from the impact of supercomputing on engineering analysis and design. Computational modeling and simulation are vitally important for the development of high-technology products in a globally competitive marketplace. Computational engineers develop and use advanced software for real-world engineering analysis and design problems.
The research work of engineering mathematics professionals and academics has potential applications across engineering disciplines. However, in the twenty-first century, mathematics is highlighted in applications surrounding environmental concerns. Engineering mathematics professionals have developed models to simulate the impact of water pressure on basal crevasses and better understand the stress profiles of grounded and ungrounded tidewater glaciers. Others have used computational intelligence techniques to optimize performance in steam turbine power plant generators. As technology, mathematics, engineering, and global needs evolve, further applications will surface.
Bibliography
Gribbin, John. The Scientists: A History of Science Told through the Lives of Its Greatest Inventors. Random House, 2003.
Koetsier, T. A History of Kinematics from Zeno to Einstein: On the Role of Motion in the Development of Mathematics. Springer, 2024.
Kumar, A., et al. "Performance Optimization of Generator in Steam Turbine Power Plants Using Computational Intelligence Techniques." Journal of Engineering Mathematics, vol. 145, no. 12, 2024. doi.org/10.1007/s10665-024-10342-6.
Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. 3rd ed., John Wiley & Sons, 2011.
Schäfer, Michael. Computational Engineering: Introduction to Numerical Methods. 2nd ed., Springer International Publishing, 2022.
Shiflet, Angela B., and George W. Shiflet. Introduction to Computational Science: Modeling and Simulation for the Sciences. 2nd ed., Princeton UP, 2014.
Stroud, K. A., and Dexter J. Booth. Engineering Mathematics. 8th ed., Bloomsbury Academic, 2021.
Velten, Kai, et al. Mathematical Modeling and Simulation: Introduction for Scientists and Engineers. Wiley-VCH, 2024.
"Women in Science, Technology, Engineering, and Mathematics (STEM) (Quick Take)." Catalyst, 4 Aug. 2020, www.catalyst.org/research/women-in-science-technology-engineering-and-mathematics-stem. Accessed 12 March 2022.