## Friday, August 17, 2012

### Odd and Even Functions - Oddly Interesting

The idea that functions can be either "odd" or "even" is an initially a surprising idea.

Yes, a number can be odd or even and some people are odd but can a function be odd or even ?

The formal definition of an even function is that f(-t) = f(t) for all of t, while for an odd function f(-t) = - f(t) for all of t.

These formal definitions don't immediately expose the power of the concept. Examining graphs of odd and even functions quickly reveal why the concept is useful and interesting. For example, below is a graph of the function is x^3sin(x), which satisfies the condition of an even function.

Figure: Plot of x^3 sin (x) (image from Wolfram Alpha)

Please note the following that the function is fully symmetrical around the y-axis (which is is the same as saying f(-x) = f(x)) As a result, iintegrating the function from 0 to any positive value of x will have the identical answer as integrating from 0 to the same negative value. This also means that integrating f(x) from x= -b to x= b is the same as doubling the definite integral from x= 0 to x= b.

Some common even functions include, f(x) = x^2,  f(x) = cos(x), f(x) = SQRT(1 + x^2), f(x) = 1/(4-x^2) and many more.

Odd functions also have an underlying symmetry but it is 180 degrees around the origin. For example, plotting  f(x) = x^2 sin(x) quickly reveals a quite different symmetry from the x^3 sin(x).

Figure: Plot of x^2 sin(x) (image from Wolfram Alpha)

For odd functions, the y value for a particular positive value of x (say x= b) will by -y for x = -b. As a result, the definite integrals from 0 to b will all be the same absolute value but the opposite sign for 0 to -b. Thus, integrating any odd function from -b to b will always results in "0" as the answer (a nice relationship to exploit when simplifying definite integrals in Fourier Analysis). This is all obvious from looking at the graph above. As often the case in mathematics, symmetry is a wonderful thing !

The relationships that follow on from these simple definitions are even also satisfying, namely:

a) An Odd function multiplied by an Odd function results in an Even function ( O x O = E)
b) An Odd function multiplied by an Even function results in an  Odd function (O x E = O)
c) An Even function multiplied by an Even function results in an Even function (E x E = E)
d) The reciprocal of a Odd function is an Odd function ( 1/O = O)
e) The reciprocal of an Even function is an Even function (1/E = E)

Somehow, these relationships seem intuitive and are in keeping with our numerical sense of "odd" and "even". They can also be easily confirmed through plotting and/or simple substitution.

In fact, I strongly encourage you to prove these relationships to yourself. That way, you will really see what is oddly interesting about symmetry.

## Saturday, August 11, 2012

### 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.

## Saturday, August 4, 2012

### 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.