Laplace Transforms on Wolfram Alpha

The use of Laplace Transforms in control theory and in signal analysis took off after WWII and has become a widely established tool of analysis. When I was a student in the dark ages (before the Internet and reality TV), we used look up tables to determine Laplace Transforms and their inverse. These tables are still widely used but online tools like Wolfram Alpha can also be readily used.

Like many aspects of Wolfram Alpha, the commands are largely intuitive and the program is forgiving with syntax:

For example, type in "Laplace Transform (t sin(2t))" into the dialogue box will produce the following answer and a graph of the function F(s). The graph of the transform is a nice bonus above the traditional look up tables, as it helps appreciate the new function that you formed through this transformation.

The inverse operations are just as simple. For example, the program only blinked for a few moments on "Inverse Laplace Transform (3s /(s^2 + 6))" to produce "3 cos(SQRT(6) t)". Once again a useful graph is generated and can be downloaded.

I encourage you to test how good Wolfram Alpha is at solving the inverse problems. Man (and Woman) against machine is always fun.

I'm sure Marquis de Laplace would have been impressed !!

Who was Laplace?

Pierre Simon Laplace (1749-1827) is a giant figure in the history of mathematics and astronomy. His career coincided with great upheaval in his home country of France, in particular the overthrow of the Bourbon monarchy and the rise and fall of Napoleon Bonaparte. Laplace was the son of a farmer (as was Newton) and was brought up in Normandy before going to Paris and becoming a Professor of Mathematics at the Ecole Miltaire. Several great mathematicians, physicists and engineers were associated with the military schools in France at the time. For example, Fourier (1768-1830) and Carnot (1796-1832) followed a similar route and had a great impact around the time of Laplace. This golden era of French mathematics and physics is directly connected to the political and social upheavals of the time. Napoleon was a great supporter of mathematics and science in France, and himself closely associated with many of the leading mathematicians of the time. Carnot's father was a general in Napoleon's army and Fourier was a trusted associate of Napoleon, famously serving as a Governor of Lower Egypt. Laplace himself is famously reported to have quipped to Napoleon "I have no need of the hypothesis", when queried by the Bonaparte why his book on planetary motions didn't mention god.

Laplace's interests in understanding the motions of planets, particularly that of Saturn and Jupiter, are linked with his developments in Mathematics . Laplace also formed a famous theory on how the planets originated. He theorized that the solar system started as a massive cloud of dust that collapsed to form the sun with the remnants condensing to form planets. An updated versions of this model of planetary formation is now widely accepted and substantial evidence has now been gathered to support this theory. Laplace came up with many novel ideas about solving differential equations, particularly, through the use of potential functions. The famous second order differential equation named after him is widely used throughout physics and mathematics. Lapalce also made significant contributions to probability theory and numerical techniques for solving equations. The famous "Laplace Transform" used widely in control theory and signal analysis is actually a variation of the approach taken by Laplace himself, though his ideas made a significant contribution  to the development of this type of analysis.

Laplace's worked during a time when there was less clear distinction made between mathematics and physics. His great success in making contributions to both fields suggest that maybe the modern tendency towards  narrow specialisation is not helpful to developing new ideas. Certainly, his work is an inspiration to those who wish to work across fields.

References

Larousse, Dictionary of Scientists, 1994

Wikipedia entry on Pierre Simon Laplace (accessed September 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.

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 !