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.
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
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
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:
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.
- "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?"
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
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.
This is by no means the end of the mathematics that is useful to engineers.
Electrical engineers should consider additional courses in:
- 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
- eigenvector, basis vector
- 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
- eix = 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
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.
- 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).
Ronald B. Standler
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posted on the Internet 13 May 2000
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