1

ON THE COMBINED TEACHING OF

ENGINEERING , PHYSICS AND MATHEMATICS

by

L. M. B. C. Campos*

Presented at Jornadas “Cultura Organizacional no Técnico”

on May 19, 2015 at IST

* Center for Aeronautical and Space Science and Technology (CCTAE/IDMEC), Instituto Superior Técnico (IST), Universidade de Lisboa, 1049-001 Lisboa Codex, Portugal.

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ON THE COMBINED TEACHING OF ENGINEERING, PHYSICS AND MATHEMATICS

A. Introduction

Title ……………………………...................................................................................................1

Index ………………………………………………................................................................. 2

B. Published volumes

1 - The series: “Mathematics and physics applied to science and technology” ………………… 3

2 - Volume 1: “Complex annalysis with applications to potential fields ……………………..… 4

3 - Volume 2: “Transcendental representations with applications to solids and fluids” ………… 6

4 - Volume 3: “Generalized calculus with applications to matter and forces” ………………..… 8

5 – Reviews of books ………………………………………………………………………... 10

6 – Comparable literature ……………………………………………….…………………… 11

C. Objectives of the course

7 – Usual sequence of engineering education …………………………………………..……. 12

8 - I. Aims for mathematics, physics and engineering ……………………………………...… 13

9 - II. Monodisciplinarsy versus multidisciplinary teaching ………………………………….. 14

10 - III. From the original sources to current ideas ………………...…………………………. 15

11 - Three methods of teaching ……….……………………………………………………... 16

12 – The student´s viewpoint ……………………………………………………………….. 17

D. 13 - Overall aim of integrated knowledge …………………………………………..…….. 18

14 - Example 1: potential fields and confocal coordinates ….………………….………..……. 19

15 - Example 2: wing sections and planforms ……………………………………………….. 20

16 - Example 3: ducts, condensers and cracks ……………………………………….…...….. 21

17 - Example 4: hodograph method and free jets …………………………………………….. 22

18 - Example 5: images in mirrors and infinite representations …………………………...….. 23

19 - Example 6: cylinders or spheres in a field …….………………………………………..... 24

20 - Example 7: non-linear bending of a beam by torques ……………………………..…….. 25

21 - Example 8: resonance of a linear oscillator ………………………………………..…….. 26

22 - Example 9: non-linear resonance, hysterisis and flutter …….…………………………..... 27

23 - Example 10: parametric resonance and excitation pass-bands …………..………...……... 28

24 - Example 11: multiple reflection of light in a lens …………………………………..…..... 29

25 - Example 12: dissipative forced wave modes ……………………………………...…….. 30

E. Conclusion …………………………………………………………………..…………… 31

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B – 1: THE SERIES “MATHEMATICS AND PHYSICS APPLIED TO SCIENCE AND

TECHNOLOGY”

Publisher: CRC Press.

Editor and Author: L.M.B.C. Campos

Published volumes:

Book A – Theory of functions and potential fields

Volume I – Complex annalysis with applications to flows and fields, 1029 pages, 2011,

(ISBN 978-1-4200-7118-4).

Volume II – Transcendental representation with applications to solids and fluids, 898 pages,

2010, (ISBN 978-1-4398-3431-2).

Volume III – Generalized calculus with applications to mather and forces, 883 pages, 2014.

(ISBN 978-1-4200-7115-3).

Book B – Bourdary and initial-value problems

Volume IV – Ordinary differential equations with applications to trajectories and oscillations

(in preparation).

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VOLUME I - COMPLEX ANALYSIS WITH APPLICATIONS TO FLOWS AND

FIELDS,

Luis Manuel Braga da Costa Campos

September 3, 2010 by CRC Press

Reference - 1029 Pages - 235 B/W Illustrations

ISBN 9781420071184 - CAT# 71181

Series: Mathematics and Physics for Science and Technology

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Features

Covers mathematical methods, physics principles, and applications in engineering and

science

Provides a coherent presentation of various areas of mathematics, physics, and

engineering that illustrates the connections among these branches

Discusses the fundamentals of complex function theory, including analytic and

multivalued functions, calculus of residues, series expansions, roots of transcendental

equations, conformal mapping, and Riemann surfaces

Incorporates applications that deal with potential flow, electro- and magnetostatics,

gravity fields, heat conduction and convection, aerodynamics, mechanical and electrical

circuits, waves, optics, and much more

Includes many illustrations, tables, and diagrams that clarify the links between topics

Summary

Complex Analysis with Applications to Flows and Fields presents the theory of functions

of a complex variable, from the complex plane to the calculus of residues to power series

to conformal mapping. The book explores numerous physical and engineering applications

concerning potential flows, the gravity field, electro- and magnetostatics, steady heat

conduction, and other problems. It provides the mathematical results to sufficiently justify

the solution of these problems, eliminating the need to consult external references.

The book is conveniently divided into four parts. In each part, the mathematical theory

appears in odd-numbered chapters while the physical and engineering applications can be

found in even-numbered chapters. Each chapter begins with an introduction or summary

and concludes with related topics. The last chapter in each section offers a collection of

many detailed examples.

This self-contained book gives the necessary mathematical background and physical

principles to build models for technological and scientific purposes. It shows how to

formulate problems, justify the solutions, and interpret the results.

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VOLUME II - TRANSCENDENTAL REPRESENTATIONS WITH

APPLICATIONS TO SOLIDS AND FLUIDS

Luis Manuel Braga da Costa Campos

April 4, 2012 by CRC Press

Reference - 898 Pages - 117 B/W Illustrations

ISBN 9781439834312 - CAT# K11546

Series: Mathematics and Physics for Science and Technology

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Features

Provides mathematical models of physical phenomena and engineering processes

particularly relevant in aerospace and mechanical engineering

Unifies interdisciplinary topics of physics, mathematics, and engineering

Explores the interplay between physical laws and mathematical methods as a basis

for modeling natural phenomena and engineering devices

Includes examples of applications with interpretation of results and discussion of

assumptions and their consequences

Enables readers to construct mathematical-physical models suited to new

observations or novel engineering devices

Contains many illustrations, tables, and diagrams that clarify the links between

topics.

Summary

Building on the author’s previous book in the series, Complex Analysis with Applications to Flows and Fields (CRC Press, 2010), Transcendental Representations with Applications to Solids and Fluids focuses on four infinite representations: series expansions, series of fractions for meromorphic functions, infinite products for functions with infinitely many zeros, and continued fractions as alternative representations. This book also continues the application of complex functions to more classes of fields, including incompressible rotational flows, compressible irrotational flows, unsteady flows, rotating flows, surface tension and capillarity, deflection of membranes under load, torsion of rods by torques, plane elasticity, and plane viscous flows. The two books together offer a complete treatment of complex analysis, showing how the elementary transcendental functions and other complex functions are applied to fluid and solid media and force fields mainly in two dimensions. The mathematical developments appear in odd-numbered chapters while the physical and engineering applications can be found in even-numbered chapters. The last chapter presents a set of detailed examples. Each chapter begins with an introduction and concludes with related topics. Written by one of the foremost authorities in aeronautical/aerospace engineering, this self-contained book gives the necessary mathematical background and physical principles to build models for technological and scientific purposes. It shows how to formulate problems, justify the solutions, and interpret the results.

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VOLUME III - GENERALIZED CALCULUS WITH APPLICATIONS TO

MATTER AND FORCES

Luis Manuel Braga de Costa Campos

April 18, 2014 by CRC Press

Reference - 885 Pages - 160 B/W Illustrations

ISBN 9781420071153 - CAT# 71157

Series: Mathematics and Physics for Science and Technology

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Features

Combines mathematical theory, physical principles, and engineering problems to

present generalized functions from an applied point of view

Examines the Heaviside unit jump and the Dirac unit impulse and its derivatives of

all orders, in one and several dimensions

Covers Gauss and Stokes’ theorems, self-adjoint and non-self-adjoint problems,

multipolar expansions, and Green’s functions

Contains step-by-step examples with interpretations of results and discussions of

assumptions and their consequences

Enables readers to construct mathematical–physical models suited to new

observations or novel engineering devices

Summary

Combining mathematical theory, physical principles, and engineering problems,Generalized Calculus with Applications to Matter and Forces examines generalized functions, including the Heaviside unit jump and the Dirac unit impulse and its derivatives of all orders, in one and several dimensions. The text introduces the two main approaches to generalized functions: (1) as a nonuniform limit of a family of ordinary functions, and (2) as a functional over a set of test functions from which properties are inherited. The second approach is developed more extensively to encompass multidimensional generalized functions whose arguments are ordinary functions of several variables. As part of a series of books for engineers and scientists exploring advanced mathematics,Generalized Calculus with Applications to Matter and Forces presents generalized functions from an applied point of view, tackling problem classes such as:

Gauss and Stokes’ theorems in the differential geometry, tensor calculus, and theory of potential fields Self-adjoint and non-self-adjoint problems for linear differential equations and

nonlinear problems with large deformations Multipolar expansions and Green’s functions for elastic strings and bars, potential

and rotational flow, electro- and magnetostatics, and more This third volume in the series Mathematics and Physics for Science and Technology is designed to complete the theory of functions and its application to potential fields, relating generalized functions to broader follow-on topics like differential equations. Featuring step-by-step examples with interpretations of results and discussions of assumptions and their consequences, Generalized Calculus with Applications to Matter and Forces enables readers to construct mathematical–physical models suited to new observations or novel engineering devices.

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5: REVIEWS OF THE BOOKS

" ... strongly application oriented. ... The language (text, figures) of the book is precise,

attractive, inspiring, and very readable. The book is strongly recommended for students,

teachers, and scientists. It is very good for self-study."

-Zentralblatt MATH 1297

“Engineers looking for detailed approaches to the use of distributions in solving problems

will certainly want to dip into this text. It certainly satisfies its stated aim..."

- MAA Reviews

"... the book will be useful to engineers who do not want to learn so much on mathematical

results but who are mostly interested by the resolution of concrete mechanical problems.

The length of the book is not a problem as the many tables may help the reader to find

quickly the solution of the problem he is looking for."

- Alain Brillard in Zentralblatt MATH

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6: COMPARABLE LITERATURE

On Mathematics:

[1] E. Goursat, Course of mathematical annalysis, 5 vols., Dover

[2] V.I. Smirnov, Course of higher mathematics, 6 vols., Pergamon Press.

On Physics:

[3] A. Sommerfeld, Lectures on theoretical physics, 6 vols., Academic Press.

[4] L.D. Landau & E. F. Lifshitz, Course of theoretical physics, 10 vols, Pergamon

Press.

On Engineering (arguably):

[ ] “Mathematics for engineers and physicists”

L.A. Pipes, C.R.Wylie, A. Bronwell, C. Lanczos, I.S. Sokolnikoff & R.M. Redhaffer, C.R.

Wylie & L.C. Barret, J. W. Deltman, K.A. Stroud, S.A. Orszag & C.M. Bender, E.

Kreysig, M. Athenborough, P.V.O´Neill, etc…

Aim: to cover all three areas art th3e same l4evel in a consistent manner.

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C – OBJECTIVES FOR THE COURSE

7: USUAL SEQUENCE OF ENGINEERING EDUCATION

The usual sequence in a 5 - year design-oriented enginnering degree with a strong

scientific background consists of:

Years 1 and 2: Basic science such as mathematics, physics, etc…

- The students ask: what is this for?

- Very abstract teaching does not help!

Year 3: Fundamentals of engineering such as aerospace, mechanical, electrical,

civil, etc...

- The student now fully understands why the basic science is so important!

- The trouble is that a part may have been forgotten in the meantime?

Years 4 and 5: Applications to real engineering problems with social and economic

relevance:

- The student realizes the limitations of his basic knowledge!

- He uses simplistic models or has to improve basic knowledge to do better?

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8: AIMS (I): FOR MATHEMATICS, PHYSICS AND ENGINEERING

For Mathematics:

a) Prove every result rigorously, with all conditions of validity and use as much as

possible intuition or images for understanding (not as proof!);

b) Break down long proofs in smaller steps, with clear intermediate results, and a

visible path through them;

c) The level of generality and the type of formalism should cover most applications

efficiently.

For Physics:

d) Present physical laws in a general form, as much as possible in a plausible and

intuitive form, not as obscure or divine inspiration;

e) Introduce one physical concept at a time, and ensure it is assimilated with examples

and applications, before going to the next;

f) Present the most general concepts, ideas and methods first in the simplest

representative context.

For Engineering:

h) State clearly all the assumptions needed to pass from the “real problem” to the

‘idealized model’.

i) Solve the model in a mathematically rigorous way, without unnecessary, redundant

or contradictory assumptions and short cuts.

j) Explain the physics of the results without having recourse to mathematical details,

and understand the pratical implications.

Conclusion: Engineering is the quantitative application (mathematics) of the laws of

nature (physics) to create new devices or services (pratical motivations).

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9: AIMS (II): MULTIDISCIPLINARY VERSUS MONODISCIPLINARY TEACHING.

There are mamy analogue problems in physics and engineering using the same

mathematical methods with different interpretations:

Example A: circuits: electrical, mechanical, acoustic networks, hydraulic networks, truss

stractures;

Example B: irrotational fields: classical gravity, electrostatics, potential flow, steady heat

conduction, etc…

Example C: solenoidal fields: magnetostatics, rotational flow, etc…

Example D: waves: acoustic, electromagnetic, elastic, water, etc…

In multidisciplinary teaching:

- Duplication is avoided: do not teach the same material with different names;

- Analogies are identified and differences must be taken into account;

- Similar methods may apply but the interpretation of results may be different;

- Reasoning in different contexts promotes interdisciplinary knowledge essential in

modern engineering.

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10: AIM: III FROM THE ORIGINAL SOURCES TO CURRENT IDEAS.

- Not just to follow the current fashion re-digested text books and consider also the original

authorities like:

- Lamb “Hydrodynamics”

- Love “Elasticity”

- Stratton “Electromagnetic theory”

- Carslaw & Jaeger “Heat conduction”

- etc…

- Find a compromise between tortuous historical development and sanitized modern short-

cuts and explain in succinct way the problem, its relevance, and the methods of solution:

- Merge the information form various fields to cover all of them without duplication, and

highlighting what is really new in each of them.

- Choose a range of examples and applications form the classical problems to modern

cutting-edge technology to show that basic principle hold longer and evolve less often

than what we use them for.

Conclusion: Being modern in applications without loosing the historic perspective of what

the original fundamental ideas were.

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11: THREE METHODS OF TEACHING

There are three ways of providing students literature supporting the lectures:

- lecture notes (preferably not too incomplete or outdated);

- choose a published textbook: plenty of goods choices of what others taught some years

ago;

- write a book: worthwhile only if the author has a different approach to the subject.

What is the difference?

- combined teaching of mathematics, physics and engineering;

- the resulting cross-fertilization: analogies and differences;

- adapting similar methods to distinct and multidisciplinary contexts.

Where does it fit:

Starting in the 3rd year when the students already know basic mathematics and physics and

understand the engineering applications, and so are open to improve their knowledge in all

3 areas.

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12: THE STUDENTS VIEWPOINT

- the subjects are “Interactive Phenomena” (1st semester) and ‘Space environment”

(2rd semester) in the Aerospace Engineering Degree.

- the subjects are optional, 3rd year for “Aircraft” and “Avionics” branches and

obligatory 4th year for the “Space” branch;

- the content changes every year and semester as the books are written and the

students receive clear typed copies of the original with figures and tables to publisher

standard.

- there is final exham with pass for a mark between 10 and 16 out of 20;

- marks above 16 are reduced to 16 unless the student goes to an oral exhamination;

- oral exhamination is possible only for marks above 16;

- the student need not take the oral exhamination but is encouraged to do so:

- the result of the oral will never be less than 16: nothing to loose;

- the student can suggest the date of the oral, that will be accepted whenever

possible, so that he can prepare himself fully;

- the oral is one or more problems that are extension(s) of the material lectured.

Final mark of oral enhamination:

- 16: student does not solve the problem(s) posed (but had good written

enhamination);

- 17: students needs considerable help to solve the problem(s);

- 18: student solves the problem(s) with moderate help;

- 19: student solves the problem(s) without any help.

- 20: student solves a problem(s) in an innovative way.

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D – 1.3: THE OVERALL AIM OF INTEGRATED KNOWLEDGE

The series is written along the following guidelines:

- the series is entirely self-contained and proves every statement (though there is an

extensive bibliography as a support);

- each chapter concerns a major topic or concept, for example “conformal mapping”

or “electrostatics” or “plane elasticity”;

- each odd-numbered theoretical or mathematical chapter is followed by an even-

numbered chapter of applications of the theory:

- the theory is rigorous with all conditions of validity and the applications are

detailed all the way to obtain practical results;

- maximum use is made of intuition, physical understanding and engineering

relevance supported on figures, diagrams and tables;

- each chapter goes beyond the standard approach or examples in textbooks,

selecting same illuminating applications in monographs;

- the final chapter deals with more fundamental, abstract or difficult topics that are

important;

- the boock concludes with a set of examples that test the understanding of concepts

and ability to use methods to solve problems in detail.

- the content is chosen for its relevance regardless of whether it ‘easy’ or ‘dificult’.

12 examples follow:

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14: EXAMPLE 1: CONFOCAL COORDINATES AND POTENTIAL FIELDS (I.36.4)

byixcosharg

2Qi

20

15: EXAMPLE 2: WING SECTIONS AND PLANFORMS (I.34.5 – I.34.6)

Transformation of plate/circle into airfoils:

Joukowski airfoil with camber

Wing of finite

span

Von Mises and Karmann-Trefftz airfoils

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16: EXAMPLE 3: DUCTS, CONDENSERS AND CRACKS (I.36.5)

Conformal mapping of a circle or half-plane into polygons with any number of finite or

infinite sides and edges (4 Schwartz-Christoffell transformations).

- Flow out of channel;

- Pitot tube to measure pressure;

- Condenser with parallel semi-infinite plates;

- Stress concentration near cracks in an elastic medium.

22

17: EXAMPLE 4: HODOGRAPH METHOD AND FREE JETS (I.38)

23

18: EXAMPLE 5: IMAGES IN MIRRORS AND INFINITE REPRESENTATIONS

(I.38.8-I.38.9)

Double row of

infinite images

potential

22

2n

1n bnz1zlog

2izf

1. infinite series

22

2n

1n bnz1z

log2izf

2. infinite roduct

complex velocity

222

n

1n bnzz2

z1

2i

dzdf

3. series of fractions

4. continued fractions (II.1)

Vortex wake

24

19: EXAMPLE 6: TWO CIRCLE AND FOUR SPHRE THEOREMS

Introduction of a circle (or cylinder) in a plane field (I.24.4 – I.24.8 – I.26.7 – I.26.8; I.28.6

– I.28.9; II.2.5 – II.2.9).

Introduction of a sphere in a spatial field (III.6.5 – III.6.9).

Electric currents flowing on a

sphere between polcs

Hill spherical vortex in a stream

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20: EXAMPLE 7: NON-LINEAR BENDING OF A BEAM BY TORQUES

Non-linear bending: large slope and deflection

Concentrated torque: xHQQxM Heaviside function

Transverse force: xQxF direction function

Shear stress x'Qxf derivative of Dirac function

Green function: x;xGdxdEI 4

4

26

21: EXAMPLE 8: ORDINARY RESONANCE OF A LINEAR OSCILLATOR (IV.2)

outside

resonance

at

resonance

near

resonance

27

22: EXAMPLE 9: NON-LINEAR RESONANCE, HYSTERESIS AND FLUTTER

-bifurcations of a non-linear oscillator

-amplitude jumps and hysteresis loop

-aeronautics: aeroelastic instability and flutter

28

23: EXAMPLE 10: PARAMETRIC RESONANCE AND EXCITATION PASS-BANDS

(IV.4)

Oscillator with vibration mounts

Waves in a periodic structures

Floquet theory and Mathieu equation

29

24: EXAMPLE 11: MULTIPLE REFLECTION OF LIGHT IN A LENS: (I.22)

Method I: Summation of series for

Reflected waves

Transmitted waves

Internal absorption

Method II: Solution of coupled system of equations

Internal upward/downward fields

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25: EXAMPLE 12: DISSIPATIVE FORCED WAVE MODES

f.kkt

Bx

Txtt

22

jij

i

forcing 7

Inertia tension stiffness damping translational spring

rotary spring

1 2 3 4 5 6

1,2 – wave equation

1,4 – diffusion equation

1,2,3 – bending waves in a plate

1,2,4 – telegraphy equation

1,2,5 – Klein-Gordon equation

2,4,5 – Schrodinger equation

Vibrations of a stressed damped elastic plate supported on translational and rotary springs.

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E – 26: CONCLUSION

The combined teaching of mathematics, physics and engineering allows the solution

of more advanced problems of pratical interest than would be possible separately.

It is a large effort with:

- a big risk: it runs counter to the increasing fragmentation of science;

- and a potential benefit: a broader more integrated view relevant to multidisciplinary

subjects.

For the student:

- provides a tailor-made up-to-date course that is not a copy of what was taught

elsewhere some years ago;

- tries to be accessible to the average student while motivating the exceptional

student, and developing the potential of both.

The lecturing of the course:

- benefits from the students reactions and questions to make the text more clear and

readable;

- makes self-study possible which is important if this large work is used as reference

for consultation.

The series serves a pair of semestral subjects with constantly evolving content. By

including all the volumes of the series It could be the basis of:

- A fourth “Mathematical and Physical Modelling” branch of Aerospace

Engineering;

- A Master in “Multidisciplinary Engineering” to follow a first degree in

Aerospace, Mechanical, Electrical, Civil and possibly other branches of

engineering.

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