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.

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.

- 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.

- First, computer programs contain mathematical relations; understanding
and fluency with manipulation of these relations
is still necessary.

Mathematics 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?"

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

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

**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

- linear, constant-coefficient, homogeneous,
ordinary differential equations of both first and second order.
**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

- 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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