Imagining Archimedes

Out of the Flatlands !

2010-08-08T17:04:00.000-07:00

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

2010-03-06T01:12:00.000-08:00

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

2010-03-03T02:18:00.000-08:00

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 !

2010-03-02T19:04:00.000-08:00

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 ?

2010-03-01T13:47:00.000-08:00

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

Fourier's great idea

2010-03-01T00:42:00.000-08:00

As students enter second year mathematics, they will be introduced to a famous mathematical concept called the "Fourier Series", which (unsurprisingly) developed by Fourier in the early part of the 19th century. The Fourier series is based on a very elegant idea that has proven to be very useful in solving equations described the motion of waves, the flow of heat and almost any function or physical behaviour that has a bit of "up and down" (which mathematicians call "periodic").

The basic idea is that any periodic function can be approximated by combining sine and cos functions in an infinite series:

e.g. f(x) = constant + a1cosx + b1sinx + a2cos2x + b2sin2x ......

In this form, the overall period of this function is 360 degrees (2 pi) - you can easily prove to yourself that when you combine trigonometric functions of different periods, the longest period dominates the overall periodic behaviour of the series. Like the Taylor series (which uses an infinite combination of polynomial terms), the more terms included in the series, the greater convergence between the series and original function.

This idea is, in fact, correct for many continuous and discontinuous functions though Fourier's original development of the series (in 1822) did not elucidate the limits of this theorem. Fourier did develop a very clever way of evaluating the constants in the equation through integrating combinations of f(x) and sine and cosine functions over one period of the function. This procedure, which now can be easily performed by computers ("Mathematica" or my favourite website http://www.alphawolfram.com/) or by a hard working second year engineering student armed with a table of standard integrals.

The Fourier series, long with Taylor's series, is one of the most important mathematical tools available to engineers and scientists for analysing wave functions (e.g. radio waves, music, surf. etc.), solving differential equations and even as a means for compressing and storing data.

The Multi-dimensional Universe

2009-08-21T21:27:00.000-07:00

In first year engineering mathematics course at Swinburne we try to lead students from the boring 2-D universe of high school (uniforms, canteen lunches and x vrs y graphs) to the much more exciting and vivid multi-dimensional world of advanced mathematics ! This progression - more of a leap - requires some imagination and determination. The first step is the visualisation of 3-D space, than a progression to a more general idea of "dimensions". When I teach partial differentiation, I get the students to think about how the slope of a hill changes when you keep one dimension constant (i.e. don't move sideways and walk upwards), as compared to the slope if you swap the dimensions you keep constant (i.e., don't move upwards but only sideways). These relationships become clearer when you draw graphs of these relationships for different physical environments (e.g. slopes in a valley as opposed to a ridge or a steep point hill). This type of approach can give you a sense of what the mathematical symbols means. These hill walking mental games are not just metaphors for the mathematical operations we are studying but direct physical examples of the mathematical ideas we are exploring.

The imagination is also required when making the next intellectual journey ....that is, seeing the concept of dimension in a more general way. For example, realising that when studying how heat is flowing through the wall of a house, you can visualise the "valleys" and "hill tops" that the temperature profile will take in the three spatial dimensions of the wall, in the same way that you extended your view of the world by moving beyond x and y graphs. If you can visualise "temperature" as an extra "dimension" in the wall than you will start de-mystifying the mathematical operations taught to you (partial derivatives, cross products, etc.). After all, we are studying these operations for largely practical reasons, such as, calculating the temperature profiles of walls, the velocity profile of gas flowing in a duct and a myriad of other engineering problems, so visualising the mathematics in physical terms provides a direct intellectual route to performing the engineering calculations that any decent engineer would like to make. Some determination is required to master the mechanics of these operations - my head still spins a little when taking the partial derivative of a partial derivative - but I would argue that the imagination/visualisation part of this trip is the most difficult and most rewarding aspect of first year mathematics.

Welcome to the multi-dimensional universe !

The Mathematics of Measurement

2009-06-27T01:49:00.000-07:00

We are surrounded by measurement devices. The modern world is abound with instruments providing values for temperature, humidity, weight, time, speed, force, pH, radioactivity, power, height, voltage, current and even our attractiveness to the opposite sex ! It is a naive person indeed who accepts a measurement on face value. Accurate and reliable measurement of any quantity is difficult and errors, whether they be random or systematic, are normal.

For example, if you are told that the temperature of your house is 26.56632 C, a thinking person would ask:

How do you know the value to such accuracy ?

At what time did you take this value and does it vary with time ?

Is the value an average of many values taken from many positions in the house or is taken from a set position in the house ?

If taken from one position, is this value representative of the "house" as a whole ?

If it is an "average" value, how precisely is this average calculated ?

Are there any corrections made for the way the thermometers are distributed in the house ?

For example, if there are ten thermometers in the basement and only one in the front lounge, wouldn't a straight averaging of these values give a distorted figure ?

How much variation is there in the values "averaged" ?

Does this variation in the case of multiple values relate to the position of the measurement or is apparently random ?

Have the thermometers been calibrated against a standard ?

These questions all converge onto two main points: What does the measurement tell us about the system we are studying and how accurate is the measurement ?

Mathematics is highly useful in evaluating many of these issues. For example, statistics can be used to evaluate variation in measurements and calculus can be used to "average" values and quantify variation. Above all, mathematics can ease the hand waving and provide quantifiable answers to these questions.

For example, imagine you are calculating the distance travelled by a trolley moving at constant velocity, using the very simple formulae:

distance (m) = velocity (m) x time (m) or D = V x t

The velocity has been measured as 22.35 m/s and the time has been measured as 10.00 s. What is the error associated with this calculation ?

Given there crude measurement, we can assume that there the random error of the measurement is one half of the last graduation of the device. Simply put, if you are using a mm graduated ruler, we can assume that the error associated with the rule is +/- 0.5 mm. This may not be correct, for example, if my sight is poor the error may become larger or if the graduations on the ruler have been badly printed, this assumption may also be too low. Another possibility is that I incorrectly placed the ruler and introduced a large systematic error (as opposed to the "random" errors I have been discussing) However, without any more information the "half the smallest graduation" principle is a reasonable starting point for our deliberations.

Therefore, in this calculation, we can assume the that the velocity is 22.35 +/- 0.005 m/s and the time was 10.00 +/- 0.005 s.

Method 1.

We know from the fundamental derivation of calculus that dD/dt is approximately equal to (small change in D/small change in t) or more simply put the gradient of the curve at a particular point is approximately equal to the ratio of a small change in the resultant variable to a small change in the independent variable. This principle can be used to approximate the error using the following formulae:

error in D = dD/dt x error in t = V x error in t = 22.35 x 0.005 = 0.11175 m.

Therefore, we have the result of 223.5 +/- 0.1 m. This approach ignores any error in the V value, as it treats the problem as being D = f(t). This would be fine if V was truly constant or the error associated with V was very small compared to that associated with t. This approach is particularly useful when the function is complex (e.g. D = Vcos (t^2)) and other methods are difficult to use.

Method 2.

We estimate the error by calculating the answer using the most pessimistic values and take this answer away from the value calculated without considering the error. In this case:

(22.355 x 10.005) - (22.35-10.00) = 223.66175 - 223.5 = 0.16175m

Therefore, the answer is 223.5 +/- 0.16 m.

Method 3.

It can be shown by a simple proof, that when the errors associated with measurements are relatively small, that when two values are multiplied together, the relative error (absolute error/value) of the new value is the sum of the relative errors of the original values. In our example, this results in:

error in D = D ((error in V/V)+(error in t/t)) = 223.5 ((0.005/22.35)+(0.005/10)) = 0.16175 m

Therefore, the answer is 223.5 +/- 0.16 m.

Clearly, the first method underestimated the error and the results from the final two techniques should be used in this case. This simple example illustrates some of the complexity in determining what a measurement really means and how mathematical approaches are useful and dealing with the complex issues associated with measurement.

The Business Mathematics Connection

2009-06-07T23:21:00.001-07:00

Niall Ferguson's "The Ascent of Money" is a highly entertaining history of business that seeks to explain how business practice has had a profound effect on human history. The title of documentary series is a deliberate pun on the influential BBC TV series "The Ascent of Man" from the 1970s. This series presented a grand overview of the history of human civilisation, in which commerce was barely mentioned, where as Greek mathematics and Galileo's trail by the church were described in great detail. Apparently, a young Niall felt that something was missing and decided that once he had become a world famous economic historian he would have his revenge ! I, for one, enjoyed the pun !

In one episode, Ferguson traced the history of lending, arguing that the fortune generated by the business innovations of the Medici family and other Italian businessman effectively funded the Renaissance. This claim may somewhat under estimate the importance of artistic and scientific ideas but is certainly an effective counterbalance to the traditional dis-taste and dis-interest that many historians have shown towards the influence of commerce on human affairs.

Of particular interest to me, was Ferguson's emphasis on the impact of the introduction of "Arabic" numerals to Europe (which we now know came from India) on the ability for traders to effectively barter and exchange currency and goods. As Ferguson explained, Roman numerals, was practically useless for large commercial transactions and that Southern European traders found the counting systems used by their counterparts from the Muslim world to be far more practical. In this way, business lead a revolution in mathematics.

This link between business and mathematical innovation is profound. The very business of counting in groups of numbers (binary, decimal or duodecimal) is almost certainly linked to the growth trade in the ancient world. The concept of exponential functions is similarly linked to the development of interest calculations and banking practices in the late middle ages. It is also well established that basic concepts of probability and statistics were developed in a business context, in particular, around the complicated calculations of insurance and risk assessment in the 19th century. This interaction between business and mathematical innovation continued in the 20th century with the development of game theory and other techniques of discrete mathematics.

I'm personally not surprised by this profound link. In my own experience in small business, the back and forward of everyday commerce is a fertile ground for innovation and new ideas. The atmosphere is very different from academia, where often new ideas can be squashed by petty snobbery's, ideological positions, intellectual fashions and just plain conservatism. In business, the attitude often is, if it works, than lets use it ! This, of course, means that lots of mediocre ideas also fly but that's part of territory.

I look forward to the next episode of Ferguson's "The Ascent of Money" and learning more about the link between "dirty money" and mathematics !

The Ascent of Freeware

2009-05-29T22:07:00.000-07:00

In the last month, a new website created by a team lead by Stephen Wolfram (http://www.alphawolfram.com/) has generated considerable interest among mathematicians, scientists, engineers and the wider community. In the popular media, the site is characterised as an attempt to challenge the supremacy of "google" but a visit to Alpha Wolfram will quickly reveal that the site offers a very different service. For example, one can type "Integrate x^2cosx" and get a full analytical answer to the integral (including the steps), an alternate solution, a graphical representation of the integral, a definite integral solution and a series expansion of the solution, within seconds. Impressive indeed ! Type in "Solve x^3 + 2x^2 + x - 6 = 0", and the full solution of the cubic with steps and graphical interpretation appear moments later. Certainly, I have been able to think of analytical problems that the software can't deal with and the on line service is not really appropriate for dealing with large data sets (see http://www.scilab.org/ for powerful freeware for manipulating matrixes and high level scientific programming), but this is nit picking - Alpha Wolfram is a triumph.

Alpha Wolfram places much of the analytical mathematical power of Mathematica and Maple in the hands of anybody with access to the web. AND IT IS FREE ! It will cause mathematics teachers at all levels to re-think what kind of homework questions are worth asking, in particular, it should push assessment towards "setting up the problem" and "analysing the answers", and away from the application of largely mechanical procedures for solving various standard equations. It maybe to early to say the traditional idea of getting a 1st year Engineering student to go through hundreds of standard integrals is now dead but certainly, this approach is in danger of becoming irrelevant and going the way of "log tables" and using Euclid's "Elements" as a textbook.

Viva La Freeware !!

In Praise of Newton-Raphson

2009-05-17T04:42:00.000-07:00

The Newton-Raphson technique for finding roots of equation via an iteration process is one of the first numerical techniques taught to students of mathematics. As a technique, it illustrates important features common to many numerical techniques used in mathematics, namely:

A) it is based on a very simple mathematical idea, that is, that extrapolating a value from a curve back to the x axis, by assuming a linear relationship, is a good way to form a more accurate guess for the intercept of the curve with the x axis,

B) after a few manual calculation using the technique, you are eternally grateful to the inventors of the computer (Hail Babbage, Turing, Zuse and friends !)

C) it is very simple to turn the procedure into an automated program,

D) the better the initial guess, the quicker you will arive at the solution and save computational time,

E) the more accurate the solution you desire, the greater the number of iterations,

F) finding a strategy for dealing with rounding errors and storing numbers with the appropriate level of precision between iterations are not trivial problems,

G) without care, it is possible to diverge of the wrong solution or (even worse) even to send the computer off to an unending loop of diverging solutions (i.e. "wrong" over and over and over again), and

H) it really works - there are few curves that it can't deal with but these are relative oddities compared to the great number of curves that the technique solves readily.

As a young engineer, I wrote several programs that used the Newton-Raphson technique to find solutions to the various equations I had formed in my models. Invariably, once I had found a good method for avoiding divergent solutions, the Newton-Raphson routine would find a solution. Like many before me, I found the technique surprisingly powerful , verstaile and useful. Now, students can "play" with the technique using graphical calculators or spreadsheet programs on a lap top. In essense, once you have a "curve", whether it be formed by data or through a known equation, the technique can be used to find solution for particular intercepts (e.g. y = 0) without having an analytical solution - that may not be possible or indeed just beyond your algebraic ability.

The Box Problem

2009-05-02T02:09:00.000-07:00

A common problem used to illustrate how differential calculus can be used for optimisation is "the box problem". The box problem goes as follows; imagine you manufacture boxes (W metres wide, D metres deep and H metres high) and you wish to minimise the amount of cardboard used to produce your standard box with volume V (V= W.D.H cubic metres).

The first step is to set up an area equation, which is the quantity that we are trying to minimize:

A = (area of the two sides defined by the width) + (area of the two sides defined by the depth) + (area of the top and bottom sides)

=2W.H + 2D.H + 2W.D

Now we have three unknowns and two equations. One option is to form solution based on an assumed ratio (C) of the width to the height, which we can use to simplify our area equation to:

A = 2W.H. + 2(V/W) + 2(V/H) by using the the volume equation to substitute for D and using C= W/H, we get :

A= 2C.H^2 + 2 (V/C.H) + 2(V/H) = 2C.H^2 + (2/H)((V/C) + V)

If we graph this function (A vrs H) and forget negative values of both A and H, we can see a clear vertical asymptote along the A = 0 and a minimum near the origin that is a function of our choices for V and C. Of course, this equation is ripe for differentiation:

dA/dH = 4C.H - (2/H^2)((V/C) + V)

At the minimum, it must follow:

dA/dH = 0 = 4.C.H - (2/H^2)((V/C) + V)

therefore,

H = ((V + VC)/(2 C^2))^1/3

Now, we have a ready way of optimising the quantity of cardboard for any given volume and ratio of height for depth. What solutions do we get if we assume a certain ratio to the width to the breadth ? Which is the true minimum (i.e. independent of our assumptions of ratios of dimensions) ? Excellent questions ! Start analysing and optimising ..... welcome to Applied Mathematics !

Practical Implications of Calculus

2009-04-24T23:30:00.000-07:00

Calculus is widely used by engineers and scientists to analyse practical problems. One common approach is to analyse a particular system using fundamental physics for a particular geometry (e.g. a force balance around a spherical particle falling in a liquid) to form equations. These equations are than either integrated or differentiated to produce useful relationships for a given set of boundary conditions (e.g. settling time of a particle as a function of size and density for a given initial particle velocity). The success of this approach normally depends on the nature of the phenomena being studied (some very chaotic and/or highly non-linear phenomena are difficult to model), the assumptions made in setting up the problems and the difficulty in solving the equations formed. Often, numerical techniques are used to find solutions to these equations and any good engineering mathematics course teaches a range of relevant numerical techniques to differentiate and/or integrate equations that are either difficult or impossible to solve directly.

Another interesting application of calculus is to analyse data. Consider a set of data collected in an experiment ..... imagine we are measuring X and Y simultaneously. When we plot X against Y, the curve generated may clearly show a relationship exists but the relationship is not simple or immediately apparent. A very simple method to start analysing this mysterious relationship, is to differentiate the X Y plot numerically (i.e. calculate the slope at points along the curve) and form a new plot of dX/dY vrs X. Now remember that we differentiate particular functions, new very specific relationships are formed. For example, differentiating a trigonometric function will generate another trigonometric function, and in the case of simple trigonometric functions like sine and cosine, functions are formed that have very specific geometric relationships to the original functions (e.g. cosine has the same shape and periodic form of sine but is "out of phase" with that relationship). In the case of polynomials, differentiating produces a function of lower order; the slope of a cubic follows a parabolic relationship, the differential of a parabolic functions produces a linear function and so on. This means that by differentiating a curve (i.e. measuring the slope of the curve at each point) some of these underlying relationships in the data maybe revealed.

This approach can be extended to differentiating the dX/dY curve formed, as double differentiation also can unlock some underlying relationship For example, differentiating sinX will form cosX and differentiating that relationship will produce a negative version of the original relationship. Double differentiation of a cubic function will generate a linear function (try it !). Thus, the "slope of the slope" can potentially tell alot about the original relationship. This line of attack can be extended to integration, through measuring the area under the Y curve and plotting this relationship against X. Of course, both taking the slope and measuring the area can be used in combination to tackle the problem.

The beauty of this methodology is that the procedure is very simple (e.g. measuring a slope of a line) and easily automated. You can try it yourself .... I suggest asking a mathematically inclined friend to dream up a complex function that is the combination of well known simple functions (e.g. cosx + x^3 + exp(x)), get him or her to form an x y table of values from this relationship and than ask you to derive the underlying relationship from this data set. The detective job in front of you is made simple by modern graphical/CAS calculators that allow ready numerical differentiation and integration of curves. Sometimes, a combination of intuition, luck and insight is required to identify the underlying relationship but the journey is normally fun. Try it !!!

Thinking about the foundations of calculus

2009-04-23T11:57:00.000-07:00

Just recently, I went through the standard derivation of the fundamental theorem of calculus with my students ..... forming tangent lines to a curve, calculating the gradient of that line using an increment, taking the increment towards infinity than repeating similar arguments for the area under a curve before forming the wonderful conclusion that the mathematics of calculating an area under a curve is the reverse of the process for calculating the gradient of a curve. In short, if you understand the mathematics of change, you also understand the mathematics of accumulation and vice versa. This was the brilliant insight that both Newton and Leibniz claimed as their own in the 17th century and formed the basis of the field we know as "Calculus".

This derivation is rightly considered one of the great mathematical breakthroughs of all time and its conclusions are indeed far reaching. During the lecture, I presented the orthodox view that Newton and Leibniz are the great intellectual heros of this breakthrough with a nod of appreciation to ancient Greeks like Archimedes who developed integral calculus via the method of exhaustion. As I was going through these arguments, I found myself questioning this idea of Newtons and Liebniz's pivotal role in the development of calculus. Wasn't the real breakthrough the idea that if you take an increment and imagine it decreasing towards infinity, you can drive useful geometrical relationships ? Isn't that idea, which I think we can accredit to Archimedes, the real intellectual breakthrough ? If you know that idea and have the tools of Cartesian co-ordinates (thank you Descartes !), than won't the relationships that Newton and Leibniz formed eventually fall out ?

Even as I write these heretical ideas down I feel my inner critic saying "No, these ideas only seem obvious because of the brilliant insights of Newton and Leibniz !" That may be true but historians of mathematics writing on calculus have shown that calculus quickly formed as a field after the developments in algebra instigated by Descartes and other mathematicis just proceeding Newton and Descartes. It is also acknowledged that Barrow (Newton's teacher at Cambridge) had an early form of differential calculus before Newton (see http://www.maths.uwa.edu.au/~schultz/3M3/L18Barrow.html for an excellent overview of his ideas). After consulting my inner critic, I think the view I am forming can be expressed as follows: understanding the importance of taking increments towards zero was a great intellectual breakthrough that allowed the development of calculus, simplifying algebra through the Cartersian co-ordinates provided wonderful tools by which to understand the mathematics of change and accumulation and the derivation of calculus by Newton and Leibniz represent the accumulation of this intellectual development. In short, their intellectual insights owe a great deal to Archimedes, Descartes and Barrow.

One of the interesting observation one can make from these discussions is that the way calculus is taught follows a very different route from its historical development. At high schools, we indoctrinate students in algebra, than introduce differential calculus and limits, and than form integral calculus. In history, calculus was formed in almost the opposite order. I suppose, as long as you understand the key intellectual points underpinning calculus, it doesn't really matter in which order you have learn't them.

A very brief history of calculators or how my brother amazed my school

2009-04-09T23:16:00.000-07:00

I started high school in 1973, three years after the end of the Beatles and a generation before the end of the cold war. Everybody wore their hair long, ludriciously wide ties were considered fashionable, most engineers (like my father) owned a slide rule and very simple electronic calculators were starting to become affordable. I remember my brother saving up several weeks of his paper round money to purchase a calculator with a square root button. The arrival of this calculator at our high school caused a sensation and my brother was asked to demonstrate this technological marvel to the headmaster. With the arrival of even more powerful devices throughout that decade, my brother and myself, and everybody else studying mathematics in the Western world, continued to be trained in the use of log tables for carrying out any calculation beyond 687 x 6578. I think the last time I used a log table Ronald Regan hadn't yet become president and computer programs were typed on cards and processed overnight.

During this time, serious letters to the papers and educational experts lamented the fall in educational standards, my year 10 geography teacher warned that global warming would see Sydney under a foot of water by 2000 and there was a general feeling with anyone over the age of 40 that using calculators was "cheating".

By the end of the 1970s and into the early 80s, calculators had advanced quickly and a range of programmable calculators were on offer. In this enlightened era, engineering students tended to be either "HP" or "Casio" adherents, though a few perverse souls identified with the reverse polish notation of the "TI" calculators. I remember quite distinctly slaving away on my Casio programmable calculator with its gigantic 2k of memory, writing quite intricate programs with the line numbering system of level 2 basic, a cute plug in ticker tape printer and an audio tape memory system. Armed with this calculating power, you felt that you could conquer the world or at least complete a pressure drop calculation for a piping system in under 10 minutes. Part of me (a very small part) still hankers for the happy chatter of my ticker tape Casio printer and the amazingly clunky graphics produced from this device. By this time, the scientific calculators familiar with modern students became standard and knowledge of the workings of a slide rule suggested either a perverted soul or a person lost in the past.

The calculator was here to stay ! My arrival in the Engineering profession coincided with the great personal computer revolution and in my own small way I lead the charge, using computer programs (now written in "high" level languages like GW Basic !!) to perform complex engineering calculations that had formerly been the province of "look up" tables and approximate solutions. Even with this shift towards computing, my scientific calculator (still a Casio man) was used on a daily basis. However, by this time my career had taken a sharp turn towards research and the graphics calculator revolution bypassed me, as I was knee deep in numerics, computational thermodynamics and writing unruly "programs" in Excel. It was only when I took my current position that I was handed my first graphics calculator. It was love at first sight ! I love the fact that I can "see" the solution of an equation, that I can calculate derivatives and integrals and even form the ABC TV symbol using parametric graphics. What is there not to love ! I even accepted the transition from being a Casio man to a TI man without suffering a nervous breakdown (OK I had a little therapy).

Interestingly, serious people are still lamenting the falling of educational standards, predicting that Sydney will be under a metre of water by ......, and most people over 40 think that using a CAS calculator is cheating.

The Continuum Assumption

2009-04-03T00:21:00.000-07:00

The engineering mathematics course at Swinburne is typical of most engineering mathematics courses around the world, in that, there is a heavy emphasis on the use of functions in analysing the physical world. In particular, there is underlying assumption (often unstated) that we can deal with physical data as a continuum (e.g. analysing radioactivity measurements using exponential functions). It is this assumption that underpins the "classical" paradign of engineering mathematics, which I would describe as:

A. analyse the physical relationships of the system being studied (e.g. force balance of a particle),

B. form equations that reflect these relationships, making appropriate simplifciations and assumptions (e.g. particle is spherical),

C. solve these equations for a given set of boundary conditions or limitations using either analytical or numerical techniques, and

D. analyse the solutions obtained against physical data, returning to first two steps if the solutions obtained are inaccurate or not credible.

This approach, and many subtle variations, has proved to be very powerful in analysing engineering problems, though complex systems where subtle changes in geometry and boundary conditions can produce large variations in behaviour (e.g. turbulence in fluids, movement of fine particles and "chaotic systems" in general) have proved difficult to model using this approach. Stephen Wolfram, in his book "A New Kind of Science" (2002) (see http://www.wolframscience.com/) argued that the classical approach was fundamentally flawed and need replacing with a new approach called "cellular automata". At the heart of Wolfram's claims was this central observation:

All of our measurements of the world are made discretely, that is, we obtain discrete numbers from our instruments (e.g. the temperature measurement from a thermometer) including our senses, and artificially impose continuous relationships upon the world by forming equations around fundamentally discrete phenomena. We could more easily, and naturally, use discrete mathematical models to describe the physical world and dispense with the classical approach.

Quite a claim !! As you imagine this book caused much debate, some of it polite and in some cases, quite inpolite ! You might find the overheads of a lecture I gave on the book interesting - see http://www.swin.edu.au/feis/mathematics/staff/gbrooks_pres.html - and there are literally hundreds of sites on the web discussing this book. You may also interested to read a much earlier (and more modest) version of the same idea by Konrad Zuse (1910-1995) who published "Computing Space" in 1967. An English translation of this pioneering work on "digital physics" is available at http://www.idsia.ch/~juergen/digitalphysics.html. Zuse was also an early pioneer in the development of the computer and was, clearly, a highly imaginative and interesting thinker.

I think the claims, details and repercussions of Wolfram's claim are a bit detailed to discuss here but I do think the first part of his central claim is uncontroversial, that it, the measurements we make of the world are discrete and the equations we impose on this discrete data reflect our intellectual choices not an underlying physical connection between equations and nature (i.e. cannon balls do not have a parabolic equation written into their structure, it is "us" that chooses a parabola to model the motion of the ball). I think this is underlying assumption to appreciate as we continue along our path of differentiating/integrating/ etc. continuous functions to describe the physical world.

My Favourite Function

2009-03-27T21:05:00.000-07:00

People have their favourite colours, football teams (go dogs !) and beaches. Why not your favourite function ?

For me there are many attractive candidates for "my favourite function". For example, I enjoy the simplicity of mx + c, the up and down of x², the surprising plateauing of x³, the lovely endless symmetry of the sinx and cosx and even the quirkiness of complex polynominals (x⁴ + x³ + x² + x). One of my associates is very fond of the hyperbolic functions but personally find their curviness rather artificial (they are just a compound of two other functions). I must admitt that I find the limited domain of most inverse functions a little off putting. Why choose a function with a limited range when you can have the whole number line !

I think looking for a favourite in any area involves the formation of various vanities and snobberies, which is what makes competitions like "the top ten albums of all time" alot of fun. It is an opportunity to laugh at your own prejudices whilst studying the quirky choices of others.

So what is my favourite ?

e^x is definitely my favourite function. Why ?

Certainly, the exponential function forms a pleasing curve but it is more its amazing characteristics that draws me to e^x. I love the fact the function is based on an irrational number but calculates something commonly observed in nature (e.g. radioactive decay, rates of chemical reactions, etc.). I find the idea that the slope of any point of the line is the value at that point (dy/dx = e^x) amazing and totally fascinating. For me, e^x is number one ! (which is only true when x = 0)

What is your favourite function ?

The End of Elegance

2009-03-19T02:56:00.000-07:00

I think there are three breakthroughs in mathematics that have really shook the foundations of the field, the first is the discovery of irrational numbers (formerly accredited to Hippaus, a member of Pythagoras's school around 500 BC but Indian mathematicans are now thought to have been earlier), the second, relates to the work of Cantor in the 19th Century in showing that infinity comes in different sizes, and, thirdly, Godel's incompleteness theorem in the first part of the 20th century, which demonstrated that attempts to form systems of axioms that are entirely logically consistent are doomed. These amazing feats of insight, intellectual rigour and imagination, initially triggered rejection and a strong counter reaction from their peers. In the case of Hippaus, legend has it, that this discovery cost him his life, as Pythagoras's followers incensed with his proof that the square root of 2 is irrational threw him into the sea ! After time, these ideas were accepted, incoporated into the mathematical mainstream and built on by thinkers who followed in the wake of these tidal waves.

Lets address the first intellectual tsunami; the discovery of irrational numbers. Why was this so important ? This discovery was important because it challenged a central notion of the type of mathematics that Pythagoras and his followers were seeking to establish. Pythagoras viewed mathematics as sacred and capable of explaining the deepest ideas and describing the natural world around them. For the school of Pythagoras, shapes and numbers were elegant expressions of profound ideas. In this intellectual climate, they assumed that numbers could always be expressed in terms of simple ratios of integers (e.g. 1/7), which they understood in geometric terms - a feature of Greek mathematics that makes it hard for modern reader to appreciate their arguments directly.

What did Happaus show ? We don't have access to the original proof but we can assume that his proof followed this type of argument:

If the SQRT (2) is rational, than is follows:

SQRT (2) = a/b where a and b are integers.

It also follows:

2 = a²/b², which can easily be turned around to 2b²= a²

We know that 2 times any number will result in an even number and that square root of any even number results in an even number, therefore, "a" must be an even number. If "a" is an even number than we can express it as 2r, where r is another integer, and we can re-arrange the equation above to:

2b² = 4r², which can be simplified to b² = 2r².

Using exactly the same argument as the one above, we can say that "b" must also be even. Now, we have a contradiction, because any ratio of two even integers can be reduced to a ratio involving an even and a odd number (e.g. 2/8 = 1/4). Therefore, it is not possible for the square root of 2 to be expressed as a ratio of two integers. In fact, this intriguing qauntity can not be directly calculated but only approximated.

This is still a somewhat shocking result. A physical representation of the square root of 2 can be easily visualised by constructing a right angle triangle with two sides the length of 1 m. The hypotenuse of the triangle must be the square root of 2 (using Pythagoras's famous theorem) .... we can see it, we can easily esimate the length using a ruler, how can it be that we can not calculate it ? This is exactly the intellectual dilemna that haunted Pythagoras, disturbed many mathematicians since the Greeks (notably Newton) and still causes one to shake your head and muse that God must be playing some elaborate joke on us. The later discovery of the irrational nature of pi and e, and Cantor's discovery that that there are many more irrational numbers than rational on the number line, just serves to deepen the shock. The type of elegance visualised by the early Greeks was over. No wonder they metaphorically shot the messenger by throwing him into the sea.

Sum of Geometric Series

2009-03-19T02:29:00.000-07:00

In our proof that repeating numbers are rational, we used the following relationship:

S = Sum of the geometric series ar^n-1
= a + ar¹ + ar² + ar³ + ar⁴......
= a/(1-r)

Where does this rather elegant and surprising relationship come from ? Certainly, this simple realtionship is rather unexpected .... why would an infinite series converge on this simple ratio ?

Like many relationships in mathematics, the proof is beautifully simple. Firstly, form the equation S - Sr = a + ar¹ + ar² + ar³ ..... - ar¹ + ar² + ar³ .... = a

Therefore, rearranging we arrive at S = a/(1-r).

QED

Are repeating numbers irrational ?

2009-03-16T20:18:00.001-07:00

The question of the nature of repeating numbers comes up when we convert fractions into binary, as even apparently simple fractions in decimal becomes an infinitely long string in binary, for example:

0.1₁₀ = 0.0001100110011 ....₂ = 0.00011₂.

On first appearances, we seem to have "changed" the type of number we are representing, just through the change of base. Have we in effect converted a rational number into an irrational number ?

No, we haven't ! This new representation of the number is still rational. The proof is as follows:

A rational number is defined as a number that can expressed as the quotient of two integers (e.g. 0.1 = 1/10).

We can express an repeating number as a geometric series:

e.g. 0.997997997997 ..... = 0.997 + 0.997 (1/1000)1 + 0.110(1/1000)2 + ...... etc.

where a = 0.997 and r = (1/1000)

It is well know that the sum of geometric series of this type = a/(1 -r), which will result in a ratio of integers (in this example 997/999).

This, because an infinitely repeating numbers sequence can be represented as a geometric series and the sum of a geometric series can expressed as a ratio of integers, such numbers must be rational.

QED (Quite easily done for non Latin speakers)

Note: Thank you to associate Sergey Suslov for his thoughts on this topic.

Floating Points

2009-03-15T22:21:00.000-07:00

How do computers deal with decimal points ? This question was asked by a student during a recent class on binary calculations.

Computer by in large use a "floating point" system which converts a number into a string multiplied by a given base to a power; mathematically,

x= f x b^e where f is a real number and e is an integer

Using this system, 110.5 in decimal becomes 1.0105 x 10². If the base is predicided and we express the real number part (called the "mantissa") as less than 1 than we can represent the number as a string of digits with the last number being the integer. Thus, 110.5 becomes 11053 in this system. Of course, computers operate in binary but the logic of the nomenclature is the same. This type of "floating point" system (and many variations) facilitate the very fast calculations performed by modern computers. Fixed point systems, which operate on a predetermined setting of the decimal or binary point are far less common. This floating point system does present problems with rounding errors and representation of irrational numbers (which have to be approximated as real numbers in this system) but still is a powerful method of storing and handling large sets of numbers.

Binary - Mathematics Made Simple

2009-03-12T10:24:00.000-07:00

Binary is counting in two. Instead of 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10, we simple count 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001 and 1010. This reduction of counting to the use of two symbols is both imagative and very powerful because it greatly faciltates the mechanisation of countng. For example imagine we have four egg cups sitting in a row on a bench. We can represent any number from 0 to 15 simple by deciding that an upside down egg cup represents "1"; so if the first two egg cups are upside down and the two remain right side, this translates to "1100" in binary and "12" in base 10.

Other mathematical operations are also simple in binary, for example, adding in binary is quite elegant, for example think of 5 + 7 perfomed in binary :

101+ 111 = 1100

This whole procedure can be reduced to a simple recipe. To get the idea, line up two rows of four egg cups (assuming your family like boiled eggs alot or you have a big family!!) and see whether you can devise a set of rules for add two numbers together. Try to do the same with eight egg cups and adding decimal numbers together .... now you can start to appreciate the power of binary.

The binary number system combined with Boolean logic (another great feat of mathematical imagination) is central to the workings of the modern computer. Counting, number manipulation and storage, can be performed with amazing speed and accuracy based on very similiar procedures to your egg cup algorithm.

Counting in tens ?

2009-03-10T15:05:00.000-07:00

Why do we count in groups of tens ? Is counting in tens the best way to count ?

Very interesting questions to consider ! Clearly, the presense of ten digits on our hands must largely explain why human count in tens because there are some clear dis-advantages with the decimal system. For example, it is difficult to count in tens using mechnical and/or electronic devices, and, in fact, computers work largely using binary (counting in twos); octal (eights) and hexadecimal (sixteens) counting systems are also used by computers and machines. "Ten" is also not as useful as duodecimal ("counting in twelves") for dividing quantities (i.e. 12 can be divided by 1, 2, 3, 4, 6 and 12, whilst 10 can only be divided by 1, 2, 5 and10). Old timers in the civilised world, and citizens of the the USA, will tell you the foot/inches are more useful than meters/centimeters for practical measurements because of this ease of division associated with duodecimal. I don't think there is any desire to move from a 24 hour clock to a 20 hour clock -I woudn't like to lose four hours of sleep ! - because of the very practical nature of the current system.

Thus, in part, we count in tens because of an accident of evolution (or the will of a god according to some). It is fun to imagine a world where we have eight fingers and thumbs and not ten. My dream of an eight finger universe can found in an earlier blog (2/1/09) entry for though inclined to entertain wild thoughts or interested in counting systems.

Note: Some societies count in fives (one handed) and others don't consider numbers past two ...... one, two and many. I'm not aware of any societies that count in 20s, which is a logical extension of counting with all available digits, but imagine how difficult remembering your times tables would be in such a society !

Torque about Cross Products

2009-03-08T19:54:00.000-07:00

Archimedes is accredited with saying "Give me a place to stand and I will move the earth". This statement reflects his profound understanding of the lever principle which lead to the invention of block and tackle pulley system, which apart from saving many backs since it's inception, represents a significant milestone in the development of machines. Levers and pulleys illustrate the mechanical advantage gained by applying a force over a distance. In vectorial form, we write the relationship as:

M = F x r = "moment" vector or "torque" provided by applying a force over a distance.

where F is the force vector and r is the postions vector, and the "x" symbol represents the operations of "cross products" or "vector products". "Torque" and "moments" are imporant mechanical concepts to understand, as they underpin our understanding of machines. The starting point is the simple lever. Appreciate how a lever works (whether it be a spanner or a see saw) and you can start to appreciate the notion of torque. A "moment" is in effect the same as "torque", except the later is generally reserved for scenario's where the moment is induced by circular motion (e.g. the torque induced in an axle by a force being applied to a wheel).

Further analysis also shows:

M = F x r = Fr sin k n

where k is the angle between F and r, F and r are the magnitudes of the two vectors, and n is the unit vector perpendicular to the plane containing F and r.

Very conveniently F x r = det( i j k, F1 F2 F3, r1 r2 r3)

(if you haven't done matrix algebra see http://en.wikipedia.org/wiki/Determinant for a quick explanation)

which is the same, as saying:

F x r = (F2r3-r2F3)i - (F1r3-r1F3)j + (F1r2-r1F2)k

where F = F1i + F2j + F3k and r = r1i + r2j + r3k

Thus, the cross product functions allows important mechanical calculations to be performed and for vectors perpendicular to a particular plane to be easily calculated.

Dot Products ?

2009-03-03T19:32:00.000-08:00

Dot products or "scaler products" are the first significant manipulation we learn to use for vectors after the famous "head to tail" rule. Unlike the "head to tail" rule of vector addition and subtraction, dot products appear initially to be somewhat obscure, however, they have both a signficant geometrical and physical meaning.

Geometrically, the definition of a dot product allows the angle between to two vectors to determined quite rapidly using:

a.b = ab cos x = xaxb + yayb + zazb

a = xai + yaj + zak and b = xbi + ybj + zbk

Re-arranging

cos x = (a.b)/(ab)

where the bold italic symbols refer to vectors, x is the angle between the two vectors and non-bold symbols refer to the magnitudes of the vector. This is a very straight forward calculation for two vectors that are defined and is much simpler than the comparable cartesian algebraic approach.

The procedure also has a physical meaning, for example, the work done (W) by a Force (F) displacing an object along a vector (r) can be calculated using:

W = F.r

In this case the dot product has a precise a physical meaning .... sounds like a good idea, simple and useful !!

Vectors: Second Thoughts

2009-03-03T15:09:00.000-08:00

In developing a vectorial algebra, we introduce the idea of unit vectors i, j and k. This initially can seem quite odd and counter intuitive - why do we need to impose "directions" onto three dimensional space ? What is wrong with x, y and z (Cartesian co-ordinates) ?

I think this type of ques ion is best answered by "doing", that is, the whole point of using unit vectors to explain relationship in space become obvious when you start to using these quantities but I will do my best to justify this choice (and like a lot of mathematics, these symbols represent an intellectual choice i.e. we choose this particular abstraction to help us develop ideas) from the beginning and independent of this experience ("a priori" is the Latin for this concept).

Lets have a go .... imagine you are interested in analysing the wind patterns over Melbourne. Your raw data is wind speed and direction data collected from weather stations dotted around Melbourne. Imagine, this data is collected continuously but "average" data is collated every five minutes. How would you represent and analyse this data ? Would you express the changes in wind direction between the various weather stations through references to their various map c0-ordinates (latitudes and longitudes, or even the Melway's grid reference system) or would you express the vectors at each location ? I think the asnwer to that question is obvious but I will let you think it through ! How would you resolve the wind speeds and directions in the areas between weather stations ? Let me again suggest that using vectors to resolve this issue will be alot easier than trying to use map based physics.

It was this type of analysis that influence physicists, mathematicians and engineers to shift to a vectorial description of the world some hundred and fifty years ago. It was simply to cumbersome to try to using a Cartesian type "map" system to analyse complex physical problems.

Vectors: First Thoughts

2009-03-01T15:09:00.000-08:00

The first topic in the Swinburne Engineering Mathematics subject for first year is Vectors. For some of the students, this will be a new topic depending on your background in high school mathematics and physics.

First things first .... what is a vector ?

A vector is quantity that has both size and direction. What does that really mean ? If I ask you how fast your car is going (assuming that you are silly enough to give your old Prof. a lift) and you answer a 100 km/hr, mathematically you have given me an answer that is described as a Speed which is a Scalar quantity (a quantity that has size but no direction). If you answer 100km/hr towards the city along the Monash freeway, you have given both direction and quantity and thus a Velocity (which is a vector).

This Monash freeway brings up some other issues about vectors because as we drive along the freeway, your instantaneous speed is likely to change as you accelerate and de-accelerate. Likewise your instantaneous direction will also change as you drive along the freeway, as the Monash doesn't follow the same compass bearing into the city, for example, it turns quite northerly near Burwood Highway but starts turning westward again as its approaches the Burnley tunnel.

In this very simple example, we can see that the velocity of the car is a function of time and position and the intuitively wise among you can see a very important topic rearing it's head, that is, CALCULUS OF VECTOR FUNCTIONS i.e the mathematics of change for functions that have both magnitude and direction. Because we are nice guys at Swinburne, we don't throw you into that topic straight away, we firstly make sure that you are very comfortable with the mathematics of vectors and the details of calculus before combining the two together. Something to look forward to !

Are vectors important to engineers ?

Absolutely, vectorial quantities are critical to engineering, they help us understand the complex stress-strain relationships in bridges, the movement of fluids in pipes and channels, the flow of air around an aircraft's wing, the interplay of electrical/magnetic fields in circuits, and numerous other examples. It is difficult to imagine engineering without vectorial analysis .... without this wonderful mathematical tool, we would be left with trying to analyse complex situations with simple addition/subtraction equations and cumbersome manipulations on X-Y co-ordinates. We would be trapped in an endless Year 10 world !! Sounds like hell to me !

How do we get good at vectors ?

The first step, in my opinion, is to be comfortable with simple physical examples before moving into the details of the algebra. This is why the problems 1 to 3 on page 12 of the student notes are important. You need to be a master of these type of problems ("A boat heads off in 20 km/hr in a NE direction with a wind blowing 50 km/hr due south ....) before moving onto the questions that are more algebraic in nature. When addressing these questions, I suggest:

a) working through the examples on page 1 to 5,

b) sketching the problem and trying visualise what the answer would look like,

c) applying the head to tail rule, being careful to distinguish between problems where (i) the resultant vector is not known (therefore, you add the two vectors head to tail) and (ii) where the resultant vector is known (there, you will need to subtract vectors to work out the vector that is missing), and

d) looking at your answer and asking yourself "Does that make sense ?".

As always, a sense of humour, determination and willingness to be challenged will help.

Historical Note:

Vectors began emerging as a distinct mathematical idea during the 19th century through the ideas of Wessell (1745-1818), Argand (1868-1822), Gauss (1777-1845) and but the first really thorough treatment of the concept is generally acredited to William Hamilton (1805-1865) who developed a form of vector algebra based on manipulating "quaternions". The function "H" which is used to express the change with time of the condition of a dynamic physical system (e.g. a set of ball flying in the air), is named in his honour. Interestingly, this development of vector algebra is tied up with the development of another important topic mathematics, that is, complex numbers. These developments are well described in the book "Unknown Quantity: A Real and Imaginary History of Algebra" by John Derbyshire (Alantic Books, London, 2006) or if you want the quick story, go to http://www.math.mcgill.ca/labute/courses/133f03/VectorHistory.html .

Welcome to 1st Year Engineering Mathematics

2009-02-26T01:04:00.000-08:00

Welcome to University life ! You have made it ! You have proven to everyone that you can pass a rigorous set of exams as entry into University - do take time to feel good about yourself. Universities are exciting and dynamic places, full of opportunity and the potential for growth. My advice is to jump right in ! Join a sporting club, try something new, form a band, introduce yourself to the person standing in the line next to you, have a debate with a Marxist anarchist and even share a joke with a Professor (a good one please !)

Inparticular, welcome to students enrolled in Engineering Mathematics 1 at Swinburne University of Technology. The first session subject is a classic introduction to Engineering Mathematics, the major topics covered are:

Vectors

Algebra

Functions and Graphs

Differentiation

Integration

The subject matter is similar to the material covered in then final years of high school but with great emphasis on applying mathematical techniques to engineering problems. As you are now at tertiary level, we are expecting you solve most of your own intellectual problems - I'm simply providing some structure to the process and , hopefully, wise guidance. You will be assessed by a combination of tests (2 worth 15%), a final exam (worth 55%) and weekly assignments (worth a total of 15%). A student notebook is available from the book room that covers all the material taught in the course and has numerous problems that will be discussed in class. The classes will be a combination of lecture and tutorial. The full details are covered in the subject description sheets handed out in first week and available on blackboard.

There are several ways you can get help with any problems you are having:

A) through discussion with your classmates (there is plenty of opportunity for that)

B) through talking directly to me in the class or by appointment (emailing before hand me is a good idea)

C) by reading the notes carefully and studying worked examples

D) by reading another textbook on the same topic (another view can sometimes help clarify a point)

E) through one on one assistance at the MASH centre, where people who are specialists in tutoring in Mathematics will work through problems with you, and

F) through looking at this blog and raising questions with me through the comments box

My general advice about studying mathematics is simple: enjoy the challenge. When you can't solve a problem, don't be annoyed, be challenged. If you are struggling to make sense of a concept, push yourself to come to grips with the problem by any means you can find. Don't settle for rote learning and memory work, be hungry for knowledge. Once again, enjoy the challenge.

What is "Engineering Mathematics" ?

2009-01-13T15:17:00.000-08:00

I teach "Engineering Mathematics". What does "Engineering Mathematics" mean ? How is engineering mathematics different from any other sort of mathematics ? Does it has an underlying philosophy or approach ?

To address these questions, we need to firstly need acknowledge that most people who teach engineering mathematics don't generally see themselves as engineering mathematicians but normally identify themselves as either a mathematican who teaches engineers, an applied mathematician or an engineer who teaches mathematics. Not surprisingly, ideas about "Engineering Mathematics" reflect these different backgrounds and perspectives, so a generally agreed defintion and overall philisophy is unlikely. I think qustions about the nature of engineering mathematics point to a related question - what is engineering ?

On that question, many books, essays and papers have been written, and hundreds of defintions provided. Here are three:

"the profession in which the knowledge of the mathematical and physical sciences gained by study, experience and practice is applied with judgement to develop ways to utilise economics, materials and forces of nature for the progressive well being of human kind"

Engineering Council for Professional Development quoted in Johnston et al., "Engineering and Society", Prentice Hall, London, 2000, p. 533.

" the art of directing the great sources of power in nature for the use and convenience of man"

British Institute of Civil Engineering 1828 quoted in Ferguson, "Engineering and the Mind's Eye", MIT Press, Cambridge (USA) 2001.

"Engineers will translate the action the dreams of humanity, traditional knowledge and the concepts of science to achieve sustainable management of the planet through the creative application of technology"

Inst. of Professional Engineers New Zealand 1993 quoted in Johnston et al., "Engineering and Societry", Prentice Hall, London, 2000, p. 533.

The issues relating to the role of mathematics in engineering are quickly apparent in these definitions. The first defintion specifically mentions the role of mathematics, the second emphasizes "art" and third has "science" and "traditional knowledge" underpinning the actions of engineers. These differences are not just reflections of people's different preferences in defining their profession but reflect much deeper divisions in underlying philosophies about engineering. For example, Eugene Ferguson argues in his book "Engineering in the Mind's Eye" that a mathematical approach in engineering design at universities has been over emphsised at the expense of visualisation and drawing. Sharon Beder suggested in her book "The New Engineer" (MacMillian, Melbourne, 1998) that the amount of mathematics in traditional engineering courses reflected the desire of early engineering academics to impress other academics in their institutions of the intellectual rigor of their programs, rather than an analysis of what level of mathematics engineers really need. Of course, there are many more who would passionality argue that mathematics is a key aspect of engineering and central to its development.

For the purposed of this discussions, I will go with the defintion provided by the Engineering Council. The key terms in the definition for me are "knowledge", "study", "experience", "practice" and "judgement". Their use in the defintion suggest that mathematics and science, combined with practical experience, help the engineer to makes judgements and choices about how to utilise resources for the benefit of humanity. This implies to me that "Engineering Mathematics" must empasize the role of mathematics in making sound choices. As I labour toward some sort of coherent defintion (please be patient !!), a few aspects of the problem are becoming clear to me:

A) Engineers need to understand how mathematical principles can be applied to practical problems.

B) Engineers need to be confortable with using mathematics as a tool to inform judgements and choices.

C) Engineers need education in fundamental aspects of mathematics, in so far as a means of developing the skills associated the applications and forming judgements on technical matters.

I think the first two points are uncontroversial (though I am often surprised what some people would like to argue with !) but the third point very much reflects a judgement I have formed from personal experience. Some people argue that education in fundamental aspects of mathematics for engineers is more about "developing thinking and intellect", others would see as a simple educational necessity (i.e. don't run before you can walk). My view is somewhat different from both these positions, that is that educating engineers in fundamantal aspects of mathematics should always be done in the context of application and techical judgements. For example, when I teach techniques for solving differential equations, I emphasise right from the start the practical implications of these techniques and the strategies that engineers use to form and solve these types of problems. I would also agree with the proposition that pure intellectual appreciation of methematics should also encouraged among engineering students (the developing thinking argument) and in fact I think that both approaches (developing thinking/learning in context) can be complementary.

These deliberations don't lead me any closer to a clean defintion of "Engineering Mathematics" but rather simple emphasize how ideas about this area of knowledge are invarably interwinned with ideas about the nature of "Engineering" - a concept itself that is subject to debate and constantly evolving. I personally find this lack of defintion and evolutionary nature invigorating.

Why does multiplying two negatives result in a positive ?

2009-01-10T01:03:00.000-08:00

Last year, a student in my first year mathematics class asked me an apparently innocent question:

Why does -5 x -5 = 25 ?

After some uncomfortable silence and a couple of minutes of moving by head around from hand to the other and scratching my chin (in a vain attempt to appear intelligent and considered)I resorted to the standard rescue line used by the intellectually challenged:

"Good question ! I'll have to think about that and get back to you."

My intellectual limitations aside, this is a good question. The other major rules of arithmetic seem obvious and intuitive, for example, if I have five dollars and you give me another five dollars, I indeed now have ten dollars (5 + 5 = 10). If you ask me to divide that amount evenly among five people, that will result in five piles of two dollars ( 10/5 = 2, 5 x 2 = 10). If from this ten dollars you take eight, I clearly have two left (10 - 8 = 2). If I owe you a further ten dollars, than it also follows that I am now eight dollars in debt (2 - 10 = -8). If I keep acumulating a debt of five dollars for five days, than I would owe twenty five dollars after five days (-5 x 5 = -25). Furthermore, if I now split this debt among twenty five people, they would each owe one dollar (-25/25 = -1).

Up to this point, these basic procedures of adding, subtracting, dividing and multilplying seem entirely consistent with our experience of the world.

However, the idea of -5 x -5 is hard to express in terms of a simple business transaction or an everyday example of counting. In fact, the idea of multiplying two negtives is quite abstract.

Do we accept that -5 x -5 = 25 by convention or is there a deeper reason ?

In fact, there are very good logical grounds why we accept that multiplying two negatives results in a positive. The argument goes as follows.

Multiplying numbers that are added (or subtracted) is the same as multiplying the numbers separately and than adding (or subtracting) the terms together (e.g. (2+5) x 6 = (2 x 6) + (5 x 6)).

Therefore, we can construct the following argument about multiplying any unknown (signified by y) with a negative:

(-1) x y = (-1) x y + y - y (yes, a bit of algebraic fiddling but true !)
= (-1 + 1)y - y (using the principle above)
= - y

Therefore, multiplying any number by a negative results in that nunmber changing signs and it follows:

-5 x -5 must produce a positive.

Logic has provided an answer where intuition and common experience have failed !

This matching of mathematical ideas with "intuition" and common experience becomes difficult as more advanced ideas of mathematics are introduced to students. The inherent abstraction associated with mathematics is one of its great attritubutes, as these abstractions lead to wonderful insights and ideas, but also can provide a barrier in learning. I think to a certain degree struggling with these ideas is a sign that you are really thinking about the issues, or at least that is my excuse !

Note: It can be argued that the distributive law of muliplication (the centre of the argument I presented) can be viewed as a convention of a particular system or rather than as a "law" and that it is possible to construct different systems built around quite different conventions. Barry Mazur explains this interesting view in his excellent book "Imagining Numbers" (Penguin, London, 2003).

Why do Engineering students study so much calculus ?

2009-01-05T13:33:00.000-08:00

Why do Engineers students study so much calculus ?

There are many answers to this one !

1. "We had to suffer now its your turn !" (grizzled old Professor)

2. "As part of a government plan to keep underdesirables (e.g. mathematics lecturers) off the street" (cynical student)

3. "So much !!!! In my day, we were doing triple integration by the time we left primary school" (mathematics Professor in early stages of dementia)

etc.

Here is my answer. Engineering students should study calculus because , firstly, it provides them with a powerful tool to analyse the physical world and , secondly, because is a beautiful topic in itself, a triumph of imagination and analysis. Differential calculus, which I would describe as the mathematics of change, is incredibly useful for analysing the movement of fluids, the flow of heat, the rates of chemical reactions, the effect of changing electrical/magnestic fields, the movements of machines and numerous other engineering examples. Integral calculus, which I would describe as the mathematics of accumulation, allows to calculate area and volumes of complex shapes, centroids and moments of structures, accumulated energy and heat, and generally the overall effect of any varying process.

The fact that both forms of calculus are directly related to each other (i.e. one is the inverse operation of the other) is also one of the most profound and useful intellectual insights of all time. For example, this means that gradient of a tangent line to parabola (i.e the instanteous rise of the curve) is 2x, given y = x². If we graph this gradient function (g = 2x), the area under that curve is the anitderivative (x²). In other words, the mathematics of change can also be used to describe the mathematics of accumulation. This is not only true for this simple example but for any function that you care to think off. Accumulation is the mirror image of change. Bravo Mr Newton and Mr Leibniz - who both worked this out in the 17th century, though there is still much debate about who was actually first to make the breakthrough (see "A Tour of the Calculus" by Berlinski for an entertaining overview of this topic).

The advent of symbolic computer programs and, more recently, CAS calculators has taken much of the pain from University calculus. This technological advance has shifted the emphasis from memorizing algebraic routines towards using calculus as a tool to analyse a problem.

Counting in eights

2009-01-02T23:39:00.000-08:00

Imagine a parallel universe where everything is the same as our world except the people occupying this alternate existence have four digits on each hand and foot, and (unsurprisingly) count in groups of eights .... how does that work ?

Lets start counting 1, 2, 3, 4 (first hand), 5, 6, 7, 10 (second hand), 11, 12, 13, 14 (first foot), 15, 16, 17, 20 (every digit has now been used).

From this little example, we can see that "16" in our base 10 world is "20" in this strange place.

How would we make sense of this world ? Lets imagine we are driving down a road in this parallel universe and we see a sign saying "Maximum Speed 65 km/hr". What does this translate to in our universe ?

Lets start with the "65" part of the problem.

65 in octal (counting in eights) = (6 x 8 ) + (5 x 1 ) = 53 in base 10

Note: 655 in octal = (6 x 64) + (5 x 8) + (5 x 1) = 429 in base 10. Converting from a another base number to base ten always follows this simple procedure. Going the other way (e.g. from a base 10 number to another base) is a little trickier but basically involves dividing the base 10 number by the base of the new number and breaking down the number into multiples of the new base.

e.g. 1 53/8 = (6 x 8) + 5 remainder = 65 in octal

e.g. 2 429/8 = (53 x 8 ) + 5 remainder = (((53/8) x 8) x 8)+ 5 remainder

= ((6) 8 + 5 remainder) x 8) + 5 remainder = (6 x 64) + (5 x 8) + (5 x 1)

= 655 in octal

Easy so far (don't ask about fractions !). What about the unit "km" ? In our system this refers to a 1000 m. Lets assume that a metre in our eight toed counter universe is the same distance as our comfortable sane base 10 world. Therefore, 1000 m in octal = (1 x 512) + (0 x 64) + (0 x 8) + (0 x 1) = 512 m in base 10.

What about the unit of hours ? Lets us assume (a common expression for mathematicians !) that an hour is the same duration of time in both universe. This means that 65 km/hr is the equivalent as 53 x 0.512 km/hr = 27 km/hr in our neck of the woods. Our parrallel universe is not a fast place; I suggest not getting out of first gear and leaving the handbrake on !

Please notice that in an octal system, numbers will tend to be "larger" i.e. need more columns of numbers. This explains why computers sometimes store very large numbers using a hexadecimal system (base 16). On the other hand, the octal system needs less symbols to express numbers ("8" and "9" are redundant in our octal universe), which taken to an extreme in the binary system ("0" and "1") reduces counting to the manipulation of on/off signals. This reduction of symbols, first envisaged by Leibniz in the 17th cenury, greatly facilitates the mechanisation of counting and, indeed, binary is used in all modern computers for the manipulation and storage of numbers. The fact that Leibniz saw this advantage some 280 years before the first computer started crunching numbers in binary, tells you something about his powers of imagination and insight. Touche Mr Leibniz (who also invented calculus in his spare time).

Imagining Archimedes

2009-01-01T20:13:00.000-08:00

Welcome to Imagining Archimedes ! This blog has two main purposes, firstly, to celebrate the beauty and power of mathematics and, secondly, to provide an online service to first year engineering students studying mathematics (the subjects I teach at Swinburne University of Technology).

The site is named in honour of Archimedes (287-211 BC), one of the most brilliant mathematicians of all time, a talented inventor and engineer, a great physicist and an inspiration to anybody who has looked at thier surroundings and imagined another world beyond mere appearances. For me, mathematics is a language of the imagination, in the same way that the best poetry, art and music try to express what we feel, see and imagine. Amazingly, mathematics is also deeply practical and has underpinned finance and technology for thousands of years. In this way, mathematics is both beautiful and powerful, and for me, Archimedes is the embodiment of this ideal.

(Please see http://www.cs.drexel.edu/~crorres/Archimedes/contents.html for more about Archimedes)

Who am I ? My name is Geoff Brooks and I am a Professor of Engineering Mathematics at Swinburne University of Technology in Melbourne (Australia). My background is in Chemical Engineering and Process Metallurgy. I started my career as a very practical engineer, designing and building all sorts of equipment in the 1980s, before catching the research bug and doing my PhD at the University of Melbourne in the field of process metallurgy. Most of my current research is focused on using mathematical models to understand and optimise the processes for making metals. The ultimate goal of this work is to minimize the impact of metal production on the environment - the original motivation for starting my PhD. My knowledge of mathematics comes from both a deep love of the topic and from the experience of using mathematics to address practical issues.

(Please see http://www.swin.edu.au/feis/mathematics/staff/gbrooks.html for more about my research and background)