Engineering Mathematics: A student’s perspective

This is an essay, rather naive, written while I was in college. You can observe the excessive use of "we" to sound more authoritative, in fact I had to get a co-author for this express purpose!

Mathematics obviously forms an important part of engineering education, since it provides us with the tools to solve engineering problems. We were never good at mathematics, but still we would like to describe our experiences of learning engineering mathematics. Perhaps it would be enlightening for us and for others to discuss such learning experiences with fellow students.

Engineering studies is unlike school education. In School math is taught as a subject for its own sake. In an Engineering course mathematics serves as the foundation on which engineering disciplines are based, so it is not here on its own right, but only to help understand the other engineering subjects. What we have seen is that the engineering mathematics syllabus of Kerala University emphasizes an abstract approach; an approach that does not makes use of physical reasoning, and keeps the mathematics isolated from the engineering subjects. The mathematics problems always deal with x and y, and never with heat and pressure or any other things which may be encountered in an engineering subject. That is, the maths course just deals with some problems and some methods to solve those problems, it does not give any indication of how to apply those methods in to a real world problem.

Take the case of Differential Equations. Differential equations are of great use in most engineering disciplines, but our engineering math course, deals with it in a way that is completely boring. The syllabus just required the students to learn many methods to solve the equations. But, before the equations can be solved, the physical problem has to be modelled and the differential equation formulated. The syllabus completely ignored the modelling aspect. Perhaps the most fun part of solving a differential equation is forming the differential equation , but we missed that entirely.

Similarly vector calculus was taught in a way that was very abstract. Physical reasoning was not used much in explaining the methods of vector calculus. Using physical insights from electrodynamics would have made it more interesting to learn and to apply, instead the syllabus just required to learn many formulas for div, curl and we didn’t have to bother what these all meant, we only had to learn how to solve the usual type of problems that came for the university exams. The situation is the same in most of other topics as well. The magic of Fourier series, that once you had learned it you get the amazing ability to write almost any function as the sum of a series of sines and cosines, was lost on us, we just saw it as some more difficult stuff to be learned to get pass in the exams.

Another issue with our engineering mathematics is that it completely ignores computer based techniques. Mathematicians may prefer to work with pencil and paper, but using computers to solve problems is very prevalent in many engineering fields. The engineering mathematics is so focused on ‘pencil-and-paper’ approach, that it ignores many interesting methods of solving problems using computers.

The problem with these paper and pencil methods is that, some of them are quite easy to do using paper and pencil for small problem sizes, but the same methods are cumbersome when implemented on computers. We are all taught to evaluate determinants. The most commonly used method is the Laplace expansion method. It seems to be the easiest method to use to evaluate determinants by hand, and we rarely have to solve determinants greater than 3x3 in class or in exams.

But, problem occurs when we have to implement a program or function to compute the determinant of a matrix on a computer. The Laplace expansion method is a very unsuited for implementing on a computer because its running time is order of the factorial of n, O(n!): for a matrix of order n, it will take a number of operations that is proportional to the factorial of n for computing the determinant using the Laplace expansion method. That is to compute the determinant of a 20x20 matrix, at the rate of one operation per nanosecond would take 20! * 10^-9 seconds. It will take several years to finish.

Of course other methods of evaluating determinants are taught too. The Gauss elimination method, converts a matrix in to triangular form by using elementary transformations, once the matrix is triangular, its determinant can be taken by taking the products of diagonal elements. This method takes only O(n^3) operations, that is it can be done by n^3 number of operations, where n is the order of the matrix. But when we are calculating on paper by hand the Laplace expansion feels much easier than the Gauss elimination, and when we are doing a computer implementation also, we are tempted to use the Laplace expansion, considering the ease of doing it by hand for small matrices. But this factor of computational efficiency was never mentioned, because the syllabus didn't care about computer implementations at all.

An approach that uses physical reasoning cannot be considered rigorous, and pure mathematics never makes use of physical reasoning. But, engineering mathematics should be closely related to the engineering disciplines; perhaps that will make it more fun to learn.

May be we were not interested enough in mathematics, may be our text books were not good enough , may be we did not listen to class carefully, but in the end we did not have as much fun with engineering math as we could possibly have had.

Ajoy W
Kiran J.L

An Evening In Bangalore

The railroad cuts through the place in a duct, the both sides of which are lined by tall and lean eucalyptuses. Beyond the trees rows of condominiums, all in drab and dull shades, but all different from each other. The single line track leads all the way to Salem, but has little traffic, barely a dozen trains a day.

They had entered into the permanent way, through a narrow path that wound by a tree. Jose and Anoop, they were both of the same height and build. They often wondered how similar they were despite having come from widely different circumstances.

Walking on the rails was difficult, since they had to watch their stride to step from one sleeper to the next once. The heath had grown up on both the sides. The rails themselves were covered with all kinds of rubbish.

They might have walked a kilometer, when there was no more buildings and roads to be seen.Even the Eucalyptus trees were gone. The thicket was tree high now. The unpolished smell of wild flowers was in the air. Then suddenly the thicket gave up and revealed a clearing, on both sides was a huge boggy pond, fringed with hyacinth, and greens on the banks, where sickly stray zebu loitered. Giant leaves of taro protruded over the water on the near side of the pond. The railroad continued over a rickety bridge with a single stone-pier in the middle.

There they sat down by the tracks, facing the opposite sides,

"Don't you have ever imagined there is a place like this, in here".

"No, I didnt".

"I always wondered what was here, if we followed down the tracks. There

is more, beyond there's a depot down the road."

"When's the next train this road".

"The last train to Dharmapuri comes down in an hour. Come, lets go down to the depot."

They started walking again, crossing precarously over the old bridge, the green water below visible through all the crevasses made by rust and decay. The thicket resumed after the bridge,on one side, while the other side the ground declines steeply just beyond the permanent way, down to the pond. The land that borders the pond might once have been rice paddies, now its just grass. The thickets give way once more as the depot approached. It was a tiny station, with just the one line, not even a loop for crossings. There was a gated road crossing at the fag end of the station.

"Lets wait here for the Dharmapuri, should be here soon".

"Ok".

They waited by the platform. Darkness spread rather quickly and the November air had become a tad more cooler. After perhaps ten minutes they could hear the horn blowing and a bright light shining from up the road. The last train to Dharmapuri was coming.

* * *

Jose took deep breaths to see if it made him better to breathe the cool Bangalore air after so long. He walked out of the airport terminal whose bland glass panels and unimaginative roof structure appeared rather unsightly to him. Outside the terminal he hired a taxi to take him to the city.It was early morning and the city was slowly waking up. The way life takes one to

places seems strange and mysterious. Sometimes when one comes back its not the same place, its a different place, a new place.

BlackBerry and the state of freedom

The central government seems to put serious efforts into needless things like trying to impose controls on BlackBerry messaging service. Government is claiming terrorists are using the BlackBerry services.So they demand that BlackBerry provide a back door for the government to snoop on its own citizens.This is outrageous and draconian.This is like being in a police state.In a free democratic country like India, it is unacceptable that the government reserves right to read the private communications of the citizens.

Incompetent zealots like Home Secretary G.K Pillai are willfully calling for the rights of our citizens for secure communications to be violated.What makes me wonder most is how the Indian media approached the episode.They were merely toeing the official line, and their reports were reading like press releases from the Home Ministry.The response of the Indian business world was appalling too: One business leader supposedly stated that BlackBerry needs to understand that business is done "differently" in India and needs to back down.

I would say it is unethical for the government to ask BlackBerry or any other service provider to provide a back door or escrow mechanism for monitoring of communications.

More ruins

If I stood up, and looked out of the window , from the room where I usually work, I can see these buildings. I've asked many people no body says anything clearly. It sits near the edge of a hyacinth covered pond fringed with heath.The buildings are not protected by any fences or palisades, and seems not to be under any care.The only people I see near it are children playing and flying kites.


Speaking of old places and kites I am remembered of Ruskin Bond's story "Kite Maker" , I do not know which place he had in mind when he wrote that story, I feel it could easily have been of this city.

The wrong shade of white

Some Indian software companies have strange dress codes. You have to "well dressed", with the tie around the neck and the shoes on the feet. The shirts have to be of particular shades. I consider infosys to be one of the biggest culprits in the regard of enforcing dress codes.

I feel it as ironic , for anybody who is a programmer , whose normal cultural stereotype would not have anything to do with being well dressed , to be forced to work wearing this sort of crap.
Besides , it is outrageous to fine somebody for not wearing the tie. These companies are getting their priorities wrong.There should be no compulsion on the people to adhere to any sort of dress code. Because being well dressed does not have anything to do with programming.

I am reminded of an incident, Andrew Tanenbaum mentioned somewhere in his writings , about the experiences he had of working for a big company. He decided to leave them when he was subject to lengthy lectures on maintaining the decorum , after he wore a shirt of the wrong shade of white to the office.

I am writing this , because a long time ago , I did not not join infosys because of the darn dress code.Hey , HR , over there at infosys, you could loose people because of the darn stupid dress code, if you are still enforcing it.

The treasures of the old city

The old city is under curfew.There are policemen in most of the streets.An outburst of communal violence have disrupted the normal flow of life in this part of the city where you are never away from the links that connect to its past , its history.

Wayside monuments , crumbling mausoleums , sarcophagi , they are all there to be seen for those who wish to see. Here I am going to describe one such historically significant construction barely a stones throw from where I live: The Paigah tombs .

The people who write guidebooks to Indian cities should be burned at the stakes.In all probability you could pick up half a dozen guide books on Hyderabad , which says Paigah tomb is a historical site where Paigahs , cronies of the old Nizams are interned.I bet none of the books ever says where they are located , and there wont be any pictures of it, leaving the reader and traveler totally clueless.In fact I could not even find any pictures of the place online, and this place is supposed to be under government care.

So by sheer luck I got to know , that this place is near where I live.Just 5 minutes walk.One Sunday afternoon , I finally decided to go an explore the place.On entry I met the guide who is also the security guard.He insisted that footwear to be removed before entering into the graveyard.

The graveyard is a simple affair with an open roof.There is some ornate decorations on top of the walls.These graves seemed to be of people of lesser stature in the Nizams court.Once you walk across the graveyard, there is an open courtyard.

There seems to be a few families living inside the complex.The guard mentions that they are caretakers.Well , to me they didn't look like caretakers , more like they were just living there.On side of the courtyard there is building with a proper roof and marble flooring.The higher ups seemed to be buried there.The guard can go on explaining these for quite a long , like this fellow who is buried here was the owner of Begum Bazar, this fellow was owner of Saifabad , and this fellow wakes up every day at 2 o clock in the night and goes to the masjid.

One you reach the end of this roofed structure , there is pond or tank behind which is a small mosque .The mosque seems to be used for active worship.Towards one side of the mosque is a dilapidated structure containing more graves.

Beyond that is another building , totally covered by vegetation and in dilapidated state.The guide refused to let me go into there.He insisted there were snakes and much more in there , and it would be positively dangerous to go in.I just think that is a place of disputed ownership with private parties , they might have gotten some restraining order to prevent visitors being brought in there.

As you travel around old city , you can see many such things.I wish if I had time and interest to go to all such places, once the old city returns to peaceful times.

Passing Out

Its been more than two years since I left college, now sitting here, perhaps I am separated from it by enough space and time, to reflect on the time I spend there, and the things I saw and learned more objectively and try to be a troll on my own blog.

Darn, no body takes college seriously. Everybody is after easy marks.Everybody wants to have some fun.Everybody wants to choose the electives which "guarantees" the most marks.There is no purpose to anything.Never mind paying hefty fees just to while time away.

yet the moment , we "pass out" of college , and enter into a job, a kind of "seriousness" comes in out of nowhere.Now everything has a purpose, everything has to be done proper.Because if it isn't so, your boss will get angry, perhaps you can even get fired for it.And that is so obscene.

When I see a person, who was an easy goer in college, turn into a sincere and true worker in a company, I feel like taking a steel bar and breaking their knees with it on the spot, because that person is the ultimate hypocrite, since he can do work for the earning of money, but he cant attach the same seriousness to college which is supposedly is done for enlightenment, and we are paying for it to boot.

The guy who does good work , only so that he earns good , and does not get fired , is a creature of small ambitions , he will probably make a lot of money , buy a good house and live happily , but I feel chances are less that he will either invent the steam engine , or write the "Ulysses".