Why Engineering Mathematics is not a "service" subject

We all have sensitive points !

I find that discussion of the 1976 Olympic Men's Hockey final distressing (Yes, Australia lost to New Zealand but please don't talk about it). A statistical analysis of the Bulldogs performance in AFL is another matter that is best left alone with me (OK, we only have one Premiership).

At work, the description of Mathematics as a "service" subject in Engineering is likely to raise the blood pressure. The term service implies that mathematics is some kind of ssecondary topic to Engineers, a kind of background material before they get to the meat of their degree.  RRRRRRR

Of course, some Engineers don't directly use much mathematics in their daily jobs, this is a particularly true in areas of management, sales/marketing and production. Even in the "hard" technical areas of Engineering such as design and research only a few are regularly performing mathematical operations in their daily jobs. Quite correctly, alot of Engineering involves "soft" skills associated with teamwork, communication and generic management skills. I feel no need to denigrate these skills compared to mathematics, physics and the core sciences associated with engineering, as it clear to me that great engineering is much as a triumph of organisation and human co-operation, as it is a celebration of powerful mathematics and science.

The film "Apollo 13" (and book by Jim Lovell and Jeffrey Kluger) is an excellent portrayal of the need for strong leadership and teamwork, as well as deep technical knowledge, in solving challenging technical problems - in this case, finding a way to safely return astronauts from a damaged spaceship. "Huston, we have a problem" is the famous catch phrase from the film. This classic understatement from Gene Kranz (the legendary NASA flight director) who muttered these words in real life, emphasised the need  for calm analysis in the face of imminent disaster. As the films shows in some detail, what follows. is a story of determination, teamwork, careful sciscientific analysis of data and systematic evaluation of the options. The hero's of the films are as much the scientists, engineers and technicians on the ground as the three men in the damaged ship. We see the various players carefully checking calculations, modifying equations and running algorithms, as the drama unfolds. Human joy is unleashed as the Astronauts voices are heard after splash down, even the rock like Kranz sheds a tear.

What a wonderful celebration of Engineers and Scientists !

And here is my point ..... all of this is underpinned and linked by mathematical skills and the language of mathematics. It is rigorous training in arithmetic, trigonometry, algebra and advanced mathematics that allows the engineers to make sound choices under extreme pressure. As they rush to find the right path home, it is confidence in the core mathematics and physics behind their calculations that allows them to make life and death decisions.

Of course, not many engineering projects are as dramatic as "Apollo 13" but the point remain the same, even when engineers are not directly carrying out mathematical operations and analysis, it is their training and confidence in mathematics and fundamental sciences that empowers them to make wise choices.  Mathematics is not only the language of technology but also one it its corner stones. Mathematics is not "servicing" Engineering, it is a core topic, a central part of its nervous system and present in all its vital organs.

In summary, my advice to any young engineer is to pay close attention to your mathematics, develop your analytical skills and avoid supporting sportings teams that have only spasmodic success.

Integration Using Wolfram Alpha

Wolfram Alpha is a web based tool that allows you carry out quite sophisicated mathematical operations, producing both analytical and graphical answers. The operations of Wolfram Alpha are based on the software "Mathematica" that was developed in the late 1980's by Stephen Wolfram and his team. The web tool is much more forgiving than the off line software in terms of syntax, though Mathematica is much more powerful and can be used as a sophisicated programming language.

However, Wolfram Alpha is simple to use, performs most mathematical opeations relevant to undergraduate and High School students and is FREE !!! 

To illustrate this great utility, head to www.wolframalpha.com, think of a mathematical operation and type in your question.

I typed in "Integrate cos(x) from x = -pi to pi" into the input box and pushed enter. If you don't feel so game as me, you can go to the example page where there are numerous examples of mathematical operations that can be performed and the input forms preferred by the program. If you guess the form, like I often do, the program will do its best to make sense of your crude mathematical jottings,

A few micro seconds later, the following output came back:

(Images from www. wolframalpha.com)

The graphs generated by Wolfram Alpha are good quality and can be saved as PDFs. I also think the graphical nature of the solution is most helpful in visualising the mathematical operation being performed. For example, the symmetry of cos(x) integral around the y axis is quite evident in the solution above. This property means that we classify cos(x) as an "even function". A parabola is another example of an even function. "y=x" and "y=sin(x)" are simple examples of functions that don't have this symmetry, which I will discuss in more detail in a later entry.

Of course, this problem is too simple, now I type in a problem that is a little more challenging:

"Integrate x^2 cos(x)"

Wolfram Alpha eats up problems like this (this one would take me several minutes using pen and paper and even a a few seconds using my well worn integral tables) and even provides the solution steps, which is an invaluable to any student (or even a rusty Professor) trying to get on top of the mysteries of integration.

and another useful graph

(Images from www.wolfamalpha.com)

Once again, note how the graphical representation of the function makes it immediately clear that the function has a particular symmetry.

Wolfam Alpha has it critics and Stephen Wolfram himself is a controversial figure but personally I am most grateful to have access to such a obviously useful tool.

Out of the Flatlands !

Today, I commence lecturing in HMS 112 (the second session subject mathematics subject for first year students at Swinburne), which is exciting for me and hopefully also for the students ! Of course, I appreciate that students returning from the break may still be building up enthusiasm for the subject ... I suggest a shot of caffeine and a gentle start.

Why is it exciting ? Because I get to escape the "flatlands" of high school mathematics for once and all; I leave behind x y plots and simple one variable problems and head towards the valleys and byways of 3-D land, where unexpected dips and rises test your powers of visualisation and calculus. Where a slope is not a simple slope but needs to be defined relative to the land around it and where the mathematical symbols get curly and more cryptic !

Even at this stage of the adventure, our powers of imagination are being tested (was is the difference between a circular paraboloid and a two sheet hyperboloid ?) and we haven't even entered complex number land yet !!!!

My advice to students entering this new land .... hold on, enjoy the ride and keep a sense of humour.

Fourier Series and other amazing feats on WolframAlpha

As occasional visitors of my blog would know, I am a great enthusiast for http://www.wolframalpha.com/. For those unfamiliar with the Wolframalpha website, it is a powerful online tool that allows you perform quite sophisticated algebraic feats with excellent graphical solutions provided with details of the algebra. Fortunately, the commands for the software are quite intuitive and easy to learn. I suggest going to the website and start playing (try "plot x^2sinx", "Integrate xcos(x^2) from x =1 to 3", "Differentiate x^2In(x)" and "Solve x^3-2x^2 + 6x - 10 =0" for starters - there is an example page to help you with syntax and common commands).

For students studying Fourier Series, the site is particularly useful. Some of the exercises I can recommends for students of the Fourier Series:

A. Visualising the periodicity of function with multiple terms

e.g. Compare a "Plot Sin(x/2) + Sin(x) + Sin(3x)" with "Plot Sin(5x) + Sin(x) + Sin(3x)"

B. Integrating terms in evaluating the coefficients of the Fourier Series

e.g. If you are determining the "a1" coefficients for f(x) = 2x + 3 over the period 2pi, "Integrate (2x+3) cos(x) from x = -pi to pi"

C. Carrying out a full Fourier expansion of f(x) over a period of 2pi for n terms

e.g. "FourierTrigSeries 2x+3, x, 6"

D. Checking whether a certain function is odd or even

e.g. "Plot x^2, sinx" to compare an even with an odd function, and "Plot x^2 sinx" to see what happens when you multiply an odd and even function.

E. Performing a half range cosine expansion of a function with a period of 2pi

e.g. "FourierCosSeries 2x+3, x, 6"

F. Performing a half range sine expansion of a function with a period of 2pi

e.g. "FourierSinSeries 2x+3, x, 6"

G. Carry our a Fourier series expansion in complex form

e.g. "FourierSeries 2x+3, x, 6"

In all cases, the software can be used to aid learning and also check the answers you are calculating or deriving, AND IT IS ABSOLUTELY FREE !

Going from words to symbols

One of the important skills that we develop in Engineering mathematics, is the ability to develop mathematical relationships from written (or verbal) descriptions. This translation (or perhaps interpretation ?) from "ideas" into equation form is challenging because its requires both mental dexterity and familiarisation with mathematical language.

It does take some confidence to translate "If you have a room of volume 60 cubic meters, that is one meter higher than it is wide and one meter longer than it is high, what are the dimensions of the room ?"

into

(x+1)(x+ 2) x = 60; w=x, h= (x+1) and L=(x+2), solve for x

This is particularly apparent when first year engineering mathematics students tackle vector problems that start with descriptions like "A ferry is crossing a river ....". I think this dis-comfort reflects a background of solving problems that either already defined in mathematical terms or has a ready made picture representation provided with the problem. Unfortunately, the problems presented to engineers and applied mathematicians are rarely presented so neatly.

My advice to developing this skill can be broken down into the following steps:

a) Try to represent the problem as a picture through a freehand sketch (labelling lines and symbols from the written description of the problem).

b) Try to visualise the problem from this picture representation, forming an image in our mind, identifying what specific problems you are trying to solve.

c) Express the problem in symbolic form, writing down definitions of the symbols you are using or any assumptions you need to make.

d) Solve the equation (or equations) you have formed, showing each step systematically.

e) Look at your answer and your original picture of the problem and ask yourself two important questions:

(i) Have you answered the original question ?

(ii) Does your answer make sense ? (Is it believable ?)

Of course, like any skill, practice will develop your abilities. It must also be admitted that there is an element of "art" to the processes described above that is beyond words or description .... which makes it fun and challenging !

Vectors made easy !

The unit vector notation used in Engineering mathematics is wonderfully simple and powerful.

Imagine we have a position in the x/y plane, lets call it P, and we want to form a vector from the origin to this point. We call that position vector OP.

Lets say that P is located at x=2 and y=3, now we can draw a line from the origin to P and put an arrow head along it. We have a vector.

Now lets define the unit vector (i.e. one unit in a particular direction) in the x direction as i and the unit vector in the y direction as j.

We can now say OP = 2i + 3j.

Lets try a few things ....

What is the magnitude of this vector ?

If we draw a right angled triangle from OP, we can quickly see that the magnitude of OP must be equal to the square root of (2 squared + 3 squared) = sqrt(13).

What is the unit vector of OP ?

It must be OP/sqrt(13) i.e. one unit in the direction of OP.

What about if we want to add another vector (OQ = 4i + 1j) to OP ?

OP + OQ = (4 + 2)i + (3 + 1)j = 6i + 4j.

What if I want to find the vector QO ?

QO = -OQ = -(4i + 1j) = -4i-1j

What about if we want to find the vector PQ ?

We use the head to tail rule and say PO + OQ = PQ = -(2i + 3j) + (4i + 1j) = 2i - 2j.

This approach makes the whole problem of manipulating vectors easy and more powerful than constantly referring to angles and scalar qauntities.

Who was Fourier ?

Joseph Fourier (1768-1830) like many brilliant scientists and mathematicians before the 20th century and modern tendency towards narrow specialisation, excelled in many fields and combined theoretical brilliance with practical ability. An orphan at the age of ten, he was educated in a military school and an abbey, showing outstanding mathematical ability from a young age. Joseph was a man of his times and played his own role in the French Revolution, subsequently serving in Napoleons armies before taking up a position at the Governor of Lower Egypt (imagine, your mathematics lecturer being the governor of lower Egypt !). He loyalty to Napoleon continued, as he also served in his armies during Napoleon's brief return to power in 1815. Certainly, the revolution had been important in providing a person like Fourier of humble birth ( he was the son of a tailor) opportunities to excel and make a mark in French society.

After several adventures in Egypt, he returned to France and mixed his ability in administration with experimental science and mathematics. He was made a Baron in 1808 and served in senior roles in the Academy of Sciences. Fourier was particularly interested in finding mathematical methods for describing heat flow. It was in this context that Fourier developed the idea that any continuous or dis-continuous function could be expressed as a infinite series of trigonometric functions. He wasn't able to prove this to be correct or general but he did develop techniques that proved invaluable since for many wide ranging mathematical problems. It is Fourier who showed that any complex wave form could be broken down into a combination of simpler wave forms. This remains a brilliant insight that guarantees his place as one of the great figure of mathematics. Merci Monsieur Fourier !

PS. In recent times, people have argued that Fourier was also the first to correctly identify the mchanism of global warming (see http://www.aip.org/history/climate/co2.htm)

Sources:

Larousse, Dictionary of Scientists, 1994, New York

Delvin, The Language of Mathematics, 1998, New York

Wikipedia