Mathematics for Physicists and Electrical Engineers

Copyright 1988 and 2000 by Ronald B. Standler

The following paper was prepared in 1988 when I was an Associate Professor of Electrical Engineering at the main campus of The Pennsylvania State University and a member of the Electrical Engineering Department's Curriculum Committee.

There are two reasons that this paper is posted here on the Internet:

Encourage mathematics teachers and professors to concentrate on certain topics, and to include derivations and "word problems" on every examination.
Inform pupils and students who are contemplating a career in engineering, chemistry, or physics that mathematics is not only important, it is essential to the practice of those professions.

cite as: Ronald B. Standler, "Editorial: Mathematics for Engineers," The Journal of Undergraduate Mathematics and Its Applications, Vol. 11, pages 1-6, Spring 1990.

Why is mathematics important to engineers? And what in mathematics is most important?

The dialogue about mathematics for engineering students must not be confined just to engineering faculty at four-year universities. Many undergraduate engineering students transfer from two-year colleges, where they took all of their mathematics classes. And all engineering students receive their mathematics "basic training" (in algebra, trigonometry, and analytic geometry) in high school.

I hope in this essay to influence mathematics instructors, particularly those who have little contact with professors of engineering, to consider the needs of their students who will be going on to engineering careers.

Three Key Reasons

There are three keys reasons why mathematics is important for engineers:

The laws of nature (e.g., Maxwell's equations for electromagnetics, Kirchhoff's Rules for circuit analysis) are mathematical expressions. Mathematics is the language of physical science and engineering.
Mathematics is more than a tool for solving problems; mathematics courses can develop intellectual maturity. It is critical that engineering students learn to visualize abstract concepts.

Many students believe that the way to solve a problem is to search for the proper formula, and then substitute numbers into the formula. This may be all right for solving quadratic equations (except when the factors are obvious), but this is not a good general attitude.

Doing derivations helps the student develop a logical thought process, a discipline of problem solving that is essential for solving engineering problems of many kinds.

Few problems can be solved immediately. It is critical that engineering students develop persistence at solving problems. Often the "best" way does not come instantly or even easily; one must try various methods and see what happens. The experience of working large numbers of homework problems, of diverse kinds, seems to build a personal collection of approaches and tools, and add to an understanding of mathematics.

Many students need more practice in how to start solving a problem, including translating "word problems" into mathematical expressions.
Numerical simulation on a digital computer is a powerful and effective tool that is being used by an increasing number of engineers. However, computers do not make traditional mathematical analysis obsolete! The following three reasons support this belief:
- First, computer programs contain mathematical relations; understanding and fluency with manipulation of these relations is still necessary.
- Second, debugging computer programs is a difficult art. One of the best ways to validate a program is to compare the computer simulation of simple situations to the analytical solution for the same situation. Knowledge of traditional mathematical analysis is essential for this method of validating computer programs.
- Third, it is relatively easy to write brute-force computer code that requires a long runtime and produces significant error, owing to accumulation of errors from the limited resolution of machine numbers. Great increases in both speed and accuracy can be obtained by using analytical solutions for parts of the problem, or by careful development of appropriate algorithms. Knowledge of traditional mathematics is highly relevant to this task.

Specific Comments on
Mathematics Courses

The following remarks concern students majoring in engineering. Students majoring in mathematics or computer science would be expected to have different needs. We hope all those needs could be met in common lower-level courses.

Here is a provocative comment, sometimes heard among professor of engineering:

"Proofs of theorems and discussion of axioms, postulates, etc. should receive minor treatment (but not be eliminated!). For example, few successful engineers are able to state Rolle's Theorem in Calculus. Is such material really critical?"

While most engineers do few, if any, proofs, they do many derivations and mathematical operations. Students should receive extensive practice in doing derivations. For example, students should know techniques for expressing sin(a+b) as a function of sin(a), sin(b), cos(a), and cos(b). Engineers usually consult tables of identities for such relations, but learning how to do such derivations is an important intellectual skill. This skill is required in courses on electromagnetic field theory, signal processing, semiconductor physics, etc. We hope that every mathematics examination includes some derivations.

There should be more unknown parameters (e.g., A, B, omega, phi) in problems and exercises, and fewer numerical values. In other words, mathematics courses should be abstract and general.

Extensive treatment of techniques for evaluation of integrals may not be a good use of time. Nearly every engineer or scientist uses tables to evaluate analytical expressions for integrals. But change of variable and integration by parts are essential techniques, even when integral tables are used.

Students need practice in algebraic manipulation to put the solution in a form that is easy to appreciate. It is difficult to state specific criteria for "easy to appreciate," but people who are fluent in mathematics seem to be able to agree that some expressions are "simpler" or more "easy to appreciate" than others. For example, it is important to be able to note easily the asymptotic behavior as the independent variable goes to zero or infinity. It is also important to be able to find the poles and zeros of a function.

Many problems in engineering and applied science involve the solution of either a quadratic equation or a set of N independent linear equations in N variables. Students should be able to solve these equations without difficulty. Some students seem to think that Cramer's rule is the best, or even the only, way to solve a set of linear equations! Gaussian reduction (Gaussian elimination) should also be taught.

The ability to write a differential equation for a particular applied problem should receive little attention in mathematics classes, as that topic is emphasized in engineering and physics classes.

Many professors of engineering and science like the treatment in Calculus and Analytic Geometry by George B. Thomas. While it is inappropriate to demand that this textbook be adopted, we urge that whatever textbook is used have a similar flavor and content.

Fundamental Topics and Goals

There is limited time in high school and the first two years of college to learn all of the mathematics that is important to scientists and engineers. A reasonable goal for engineering students by the end of their sophomore year is to be able to solve the following types of problems.

Algebra
- factor expressions
- solve N independent linear equations with N variables
- find roots of any quadratic equation
- use logarithms to multiply and divide
- use logarithms to raise any number to any power
- change of base of logarithms
- arithmetic and geometric series
- matrices
- eigenvector, basis vector
Calculus
- take a derivative of any function
- integrate functions of one variable (may use tables)
- find maxima and minima of a function
- partial derivatives
- find mean value of any function (moments, centroid)
- find the derivative of integral whose limits are functions of the variable
Analytic Geometry and Calculus
- trigonometric identities
- law of cosines, solution of right triangle
- coordinate systems: rectangular, cylindrical, spherical (and also their two-dimensional counterparts)
- equation of a straight line: point-slope, slope-intercept, etc.
- equations for conic sections
- transformation of coordinates
- distance between two points, point and line, point and surface
- find surface area
- integral of a function over a surface
- find volume
- integral of a function in a volume
Vector calculus
- arithmetic with scalars and vectors, unit vectors
- dot product of two vectors (scalar product)
- cross product of two vectors (vector product)
- linear independence or dependence of two vectors
- divergence and curl operations
- find tangent and normal vectors (to a curve or surface)
- how to apply Divergence and Green's theorems
Differential equations
- linear, constant-coefficient, homogeneous, ordinary differential equations of both first and second order.
- same, but non-homogeneous with an inhomogeneous part that is either:
  A + B t
  A cos(omega t + phi),
  A exp(-t/tau), or
  A exp(-t/tau) cos(omega t + phi).
- examination of response in both the time and frequency domains
- resonance and damping (underdamped, critically damped, overdamped)
- methods of undetermined coefficients and the Laplace transform
Complex numbers
- Argand plane
- a + ib = c + id if both a = c and b = d
- e^ix = cos(x) + i sin(x)
Infinite series
- how to find Taylor's series expansion (and how and when to use it)
- Fourier series
- tests for convergence of series

This is by no means the end of the mathematics that is useful to engineers. Electrical engineers should consider additional courses in:

ordinary differential equations,
partial differential equations (boundary value problems),
functions of a complex variable,
statistics, and
numerical analysis (particularly: root finding, integration, evaluation of transcendental functions, fitting equations to empirical data, numerical solution of differential equations).

I doubt that many traditionally educated mathematicians will balk at any of the above. In an era, however, when the content of the core mathematics courses of the first two years of college is up for free discussion, engineering faculty feel it is important that the needs of the engineering curriculum be heard.

Ronald B. Standler