Differentiation - Demystified

All of us might have studied differentiation. But how many of us like this?
The moment we hear differentiation most of us will remember about d/dx. In school days we used to byheart the differential of many terms and use it for calculations.

What is this differentiation all about, why it is needed and what is it practical application?

Let me try to demystify DIFFERENTIATION through the following questions and answers. Do read it and let me know whether this helped to make your understanding of differentiation more clear.

Is there any relation between differentiation and difference?
Indeed yes, differentiation is infact derived from difference. The relationship between these two will be evident from some of the other questions here.

Why is differentiation needed in real world?
Assume that a huge block of ice is melting. Depending on the atmospheric temperature the rate at which ice melts will vary. Can we compute the time required for the melting of ice without actually waiting for the whole ice block to melt.

Suppose we are heating one end of a long iron rod. Can we compute the heat the other end of the iron rod without actually measuring the temperature at the other end, or can we measure the temperature at any given point of the rod?

A rocket needs to be launched. Can we compute where it will be, once the accelaration of the rocket is known.
All the above computations are possible if we know differentiation.

What is differentiation?

Differentiation represents rate of change of any quantity with respect to another quantity.

What does the above sentence means?

Consider the following table, quantity1 and quantity 2 is varying as shown below

quantity1	quantity2
1	2
2	4
3	6
4	8
5	10

Quantity1 is changing by a unit of 1. (How did I get this - -> take the difference of adjacent values)
Quantity2 is changing by a unit of 2. (How did I get this --. take the difference between adjacent values)

What is the rate of change of quantity2 with respect to quantity1 or

if the question is phrased in another manner
How much will quantity2 change when quantity1's value is changed by a unit value?
To find this take any two adjacent values of quantity1 and its corresponding values in quantity2.
Let us take the first two values -> quanity1 - 1 & 2, quantity2 -> 2 & 4
Rate of change of quantity2 with respect to quanity1 = (4 - 2)/(2 - 1) = 2;
take the next set of values, computing similarly we will get = ( 6 - 4 ) / ( 3 - 2 ) = 2

If you take any set of value, you can see the rate of change is 2.
Even though the values are changing, the rate of change is a constant here.

So for the above set of quantities we found out the rate of change.

This rate of change is what is called differential.

Here we found out the differential by taking the ratio of difference of two sets of values.

In mathematics, quantity2 can be represented as a function of quantity1.
Watch the set of numbers of quanity2 and there corresponding numbers in quantity1.
it can easily be found out that all the values satisfied the following relation:

quantity2 = 2 x quantity1, if quantity2 is denoted by 'y' and quantity1 by 'x', then we can say that

y = 2*x

We had found out differential of quantity 2 with respect to quantity1,

ie differential of y with respect to x.

This is denoted by dy/dx.

This means rate of change of y with respect to x.

Here it was shown that when y = 2x, the differential - dy/dx = 2.

Another example of differentiation

consider the function y = 2,

this means that the value of y will be 2, whatever be the value of x,

so our earlier table will look like this

x	y
1	2
2	2
3	2
4	2
5	2

dy/dx = (2-2)/(2 - 1) = 0

Take any pair of values, the differential of y with respect to x will be zero.

This means that when y = 2, dy/dx = 0.

The examples given above shows that, whenever one quantity varies its value with respect to another the rate of change of quantity with respect to another quantity can also be represented as a function.

for the function y = 2x

differential is 2,

for the function y = 2

differential is 0

The underlying principle of differentiation shows that the difference in values of one quantity with respect to another is called differentiation.

Purpose of taking differentiation can be summarized as follows

Most of the real world problems can be expressed as functions.

To examine the rate of change of a function, we have to take the differential of the function.

The resultant equation after differentiation gives an expression which corresponds to rate of change of the function.

There may be cases where we take the differential of the result of differential of a function.

This is needed to find out the rate of change of "rate of change" of a function.

y = f(x) -> y is a function of x, ie y varies when x varies according to the relation.

dy/dx = d f(x)/dy - differential of y with respect to x, gives the rate of change of y with respect to x.

Once this is computed the resultant differential can be used to find out the rate of change of y for any given value of x, easily.

How will differentiation help in finding out the heat at a particular distance of iron rod whose one end is heated?

Let us assume that we know the equation for heat transfer as a function of distance from the heat source.

Differentiate this equation with respect to distance.

The result of this differentiation will the equation corresponding to the rate of change of heat at any given distance from the source.

Using this equation we can easily compute the heat at at particular point away from the heat source.

Hence differentiation of a function helps us to realise equations corresponding to the rate of change of the function. The resultant equations can be used to solve pratical problems concerned with rate changes.

Engineering Mathematics - Demystified Part 2

Continued from part 1
Is mathematics a science or is science a part of mathematics?
Does anyone remember the advertisement of "Coffee Bite". There is an argument on whether the toffee is a coffee or a toffee and the punch line is that "the argument continues".The same hold good here.
Meaning of science is "field of knowledge". In this sense Mathematics is part of science.
But science deals with only physical world. In that sense mathematics cannot be part of science since there are mathematicians who simply derive certain theorems without any practical application in mind. These theorems may or may not turn into applications at a later stage.
So let the argument continue. Let us don't worry too much about that. Just know the above two facts.

What is the meaning of Mathematics?
Mathematics is derived from a Greek word - mathema, which means learning, study or science.
Is the usage "Maths" correct?Maths is simply a short form of Mathematics.

What is the definition of mathematics, How it evolved?
The standard definition goes like this
"Mathematics is the study of
1. Quantity
2. Space
3. Change
4. Structure"
Now we can try to analyse this defintion in a simple manner.
Quantity - Means study of numbers. Only if number exist we can measure quantity. All the operations done on the numbers are called "Arithmetic".Branch of mathematics which deals with study of quantity is called ARITHMETIC. Study of quantity is the fundamental in mathematics.
Space - Space means study of dimensions. We have one dimension (1D), 2 dimension (2D) and 3 dimension (3D). Dimension means a measurable quantity. 1 dimension means one measurable quantity, 2 dimension means 2 measurable quantities Why dimension is needed? To represent objects. Once we appreciate the concept of quantity, the same is being used to model objects in space. It is in this context we study trigonometry and geometry.Hence mathematics is a study of space.
Change -Some "quantities" change their value with respect to time. When a quantity changes with respect to time, we call it a dynamic system. In real world there are many examples of dynamic system vizwhen we boil water the temperature of the water increasesthe speed of a vehicle etcThe study of such dynamic systems comprises one area of mathematics. Hence mathematics is a study of change.

Structure -It was mentioned that by space we meant the study of objects having different dimensions. Each of these objects have an internal structure. The study of these internal structure is also part of mathematics. A detailed discussion on this will only confuse things. Hence I am limiting this here.
From the above list (ie quantity, space, change and structure), we can easily pick the one which must be evolved initially.
Yes it is quantity.
Now how did this evolve?
Man knew that apples are different from oranges.
But when he saw Three apples and three oranges, he recognized that there is something in common.
The commonality is quantity.
This realisation of quantity was the first step in evolution of mathematics.

Some interesting quotes on mathematics
"Mathematics is the queen of sciences" - Gauss
"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality" - Albert Einstein
"Mathematics is the science that draws necessary conclusions"

New Topics to come later about Mathematics
Differentiation - A practical description
Matrix - What is it and why it is needed?

Engineering Mathematics - Demystified (Part 1)

Everyone studies engineering mathematics upto sixth semester or so and completes syllabus equivalent to mathematics degree course, but do you know the following facts?

Why an engineer needs to have a mathematical base?
First of all who is an engineer.

A person who uses scientific knowledge to solve practical problems

So an engineer needs to have scientific knowledge.
Applying scientific knowledge means, the solution defined by an engineer should have a theoretical base. ie it should not be arbitrary solution just like we answer multiple choice questions.

How to prove this theoretical base for the solution arrived at.

One way is to express the problem in a mathematical way. If expressed mathematically, then a solution can be arrived at using mathematical equations. (But this is not easy)

Hence all engineers need to have a mathematical base.
But by learning the entire mathematics syllabus and getting high marks in mathematics exam does not imply that the engineer has the ability to express everything mathematically and understand the practical applications of every mathematical theorem which he learns.

For this, the person need to understand the concept the mathematics in relation with the real world. Then it is assured that the person will be able to atleast understand the theoretical solution put forward by the hardcore mathematicians and apply it for solving engineering problems.

Hence, if an engineer has a mathematical base he can- model the practical problems mathematically and arrive at a solution- use existing mathematical solutions available in theory and apply it in solving practical problems.

The problem here is that mathematics tutors are generally the one having only the mathematical base. It would be really good and benificial for the students if an engineer who has the experience of following a mathematical approach for problem solving, deliver atleast some lectures during the course (this never happens, in reality)

What are some of the applications where an engineer need to apply mathematics?

Let us take a very simple example where the application of mathematics comes in as part of our daily chore.

Example 1:

Assume Nandu is having a bag full of apples. He needs to divide it equally among 4 of his friends. This is the problem statement.

An aribitrary solution

If he solves this arbitarily, what will Nandu do?

He will take out a handful of apples and give to each of his friends. But can he be sure that each one of them got exactly the same number of apples. Arbitariness means result is not guaranteed.

Mathematical solution

The mathematical approach will be to count the apples, divide the value by 4 and get the result, say 'n'. Take out 'n' number of apples and give it to friends. Since a mathematical theory was put to use, it is assured that the result is guaranteed.
This example, eventhough quite simple clearly demonstrates the concept of applying mathematical theory to solve practical problems. Now why did we feel that this is quite trivial. Right from our childhood we are practising this kind of problems and the concept of division along with its practical application is clearly etched in our mind.

Similarly once a theorem is clearly understood in relation with it practical application, it will sound very simple.

Example 2:

Mathematics theorems are quite useful in deriving solutions. The popular among them is transform.
Now what is a transform?

Transforms dictionary meaning is to change something. It has the same meaning here also.

Assume transform as a kind of magic box. A set of numbers are given as input to this box. The output will contain another set of numbers, but their values might have changed. But the Number of inputs will be equal to number of outputs. So some transformation occured to input numbers and the transformed numbers are obtained as output.The magic box might contain some formula to operate upon each of the numbers and generate the output.

So why transform is needed?

The answer in simple terms is to find solutions easily.
Here is an example.

Assume I want to find the sum of any given number and 99.

I need to do this in mental calculation.
the situation is
y = x + 99
For doing this easily, I am going to transform the input number.
y = (x - 1) + (99 + 1)
y = (x - 1) + 100
The transformation applied here, subtracting the input number by 1, and adding the fixed input by 1.
I want to add 25, 27 and 58 with 99 and get the result. Without applying transform, if I add then I have to compute 25 + 99, 27 + 99 and 58 + 99 which is little bit difficult
If transform is applied, the input numbers become, 24, 26 and 57.
Add these numbers with 100 which is quite simple
the result is 124, 126 and 157.
Now we got an effect of transforming numbers. Here the computation becomes easier.
Similarly there is another transform called Fourier transform. This is also an equation to convert one set of numbers to another set. The transformation is done for making certain computations easy. I will deal with fourier transform in detail later.

Questions to be dealt with in part 2

Is mathematics a science or is science a part of mathematics?
What is the meaning of Mathematics?
Is the usage "Maths" correct?
What is the definition of mathematics, How it evolved?

Engineering Graphics - Demystified - Part 2 - Applications

What are the steps in creating an engineering drawing?

In engineering graphics questions related to lines and solids will be like follows

1. A line is at a distance of x cm from the horizontal plane and y cm from the vertical plane. It is tilted by some angle from the horizontal and vertical plane.

2. A solid (cylinder, prism etc) is at certain specific distance from both the planes and it is tilted by a certain angle.

Once you see such a question you shoule be able to think in the following lines,

• close your eyes
• think about the shape
• Think about a horizontal and vertical wall
• Now think about placing the object parallel to both the walls.
• Now imagine tilting the object in one direction (tilting from the vertical wall)
• Now you see a 3 dimensional view of the object in your mind, which is tilted in one direction
• Now imagine tilting this object in the other direction also.
• Now visualize as if you are seeing the object from the front. If possible draw a free hand drawing of what you visualized in a rough paper
• Now visualize as if you are seeeing the object from the top. Draw a free hand drawing of the top view.
• Open your eyes, follow the procedures given in the text book or as taught by your instructor and Draw the pic.
• During each step just think about the things you imagined earlier.
• Finally when you get the final top view and front view, cross check it with the rough drawing.
• if there is an error againg try to visualize the whole thing. .

If you learn like this, then engineering graphics will achieve its intended objective.

Where will you be applying engineering graphics in real world situations?

Once you are in a job as an engineer, all the basic tools and fundamental understanding for solving the real world problems are expected to be available in you.

Engineering graphics is one such tool which will aid to solve some of the problems in hand. for a mechanical engineer, civil engineer, production engineer, architect etc there is no question about where it will be used.

For students of other branches it may come handy in the following situations.

There will be situations in your job, where you have

• to visualise a housing for the electronic ciruit module which you designed,
• to design a electronic circuit board for a prescribed housing, the housing diagram will be given to you as an engineering drawing,
• For a system analyst, it will aid in drawing real world objects while creating a system drawing with real world objects (concept of perspective projection is useful here). A system drawing with real world objects will help to convey the problem solution to the user very clearly.

An excellent tutorial for learning engineering graphics, especially orthographic Projection is here

http://www.ul.ie/~rynnet/orthographic_projection_fyp/webpages/what_is_ortho.html

Engineering Graphics - Demystified

Whichever branch we opt for in Engineering, we have to study engineering graphics.

For most of the pupil this is a subject which is tough to score marks.

The teaching method of graphics is as follows (followed by most of the tutors)
Teacher will explain the rules to draw a top view or front view
Students will by heart the rules and learn to draw using T square

The end result -> Students will never get answers to following questions.

1. What is graphics?
Graphics is the representation of objects as images. In this representation the dimensionals details or the size of the object will also be quite evident.

2. Why graphics is needed?
Two people can communicate clearly only if they know the same language. Even then quite often there can be communication gap, which means that the listener might have understood things in a different way than the speaker indented.
If two persons don't know the same language then how can the speaker tell the listener that he is telling something about a car. The speaker can show the image of car to the listener.
Yes images have no barriers. There is no language for images. If you can see images then it conveys everything clearly.
But by just showing image of a table, can a carpenter make a table?
No, Because he don't have information on what size it should.

So the image need to contain size (dimensional) details also in a specific unit. Seeing this image the maker of the table can make the table exactly of the same size which is intended.

"GRAPHICS IS A UNIVERSAL LANGUAGE FOR ENGINEERS"

Why front view/ top view side view is needed?
A rectangle can easily be represented in a single image showing the length of each side. This is two dimensional figure, since it has only two dimensions, the length and breadth.
Now how can we draw a rectangular prism in a sheet of paper. Rectangular prism means it has got length, breadth and height.
Assume that we place the prism on a flat surface. Now look at it from the top in such a way that we are not able to see its sides. We can see only a rectangle now. It is possible to draw the rectangle showing its length and width on a paper. This means from the top the original 3D object looked like a 2D object, which can be represented in a paper.
Now look the prism from the front in such a way that we cannot see the top. We can see another rectangle now. This can also be drawn on a paper.
Ok now we have two images for a single object. We have to draw it in certain order according to standard conventions. Otherwise how can a third person know which is the view from the top and which is the view from the front.

More questions related to graphics to follow

Kerala Engineering Admission - A routine event done by Parents for their wards

Engineering - the course to mould up engineers is the aim of most of the students who have completed 12th grade. It has now become an ever similar routine for the students to go to one of the professional colleges after 12th just like going to 10th grade after 9th.

The emergence of lot of private engineering colleges has made life easy for students.

The attraction is that students are in for earning more than 2 to 3 lakhs per annum within the next four years or so. So naturally parents of this side of country who are naturally having a big clout on their wards, want to complete their responsibility by getting them into one of the engineering colleges.

Parents - the community consists of lower middle, middle, upper middle class working folk, teachers, businessmen every one want his child to get into a professional course. They may or may not be familiar about the different branches.

TV programmes/ Newspapers/ Counselling agents websites flood with information on which branch is good and which branch is bad. Parents reach on conclusion based on these information. They discuss these matters with parents of other students and then given options.

During all these stages the students ambition is not taken into account. To make matters worse, if one asks the student which branch to study they are clueless. They also make decisions based on popular demand.

The reason is the lack of proper technical training in schools upto the twelth grade where only theory is important.
The students have to just learn a lot of stuff byheart and a students ability to remember things is tested during exams rather than his other technical or logic skills.

This system is bound to continue and is still continuing.
What can best be done in this situation is something which the parents can do.
Some suggestions are

1. Encourage self learning of their wards from a very young age
2. Find lot of time to talk with your kids. For this to be practical parents should stop watching TV during prime time.
3. Discuss the subjects which his ward is studying in the current academic year during informal discussions. This can be done from a very young age.
4. Encourage the student to take up some activity related to the what he is studying. If this is done, students will land up in taking up small projects related to the subject which is of more interest to him.
5. Don't give complete help or do projects for your wards. Quite often parents tends to do the project work given to students at school to enable him to get more marks.
6. Watch your wards progress very closely but don't interfere with his activities or don't overly guide him.
7. Ensure that parents have passed on the responsibility of finishing the lessons to his wards. Most of the time parents take this responsibility and ends up beating/screwing up his ward to finish of his lessons.
8. Never ever complain or make bad remarks about the school/teachers where the child is studying in front of the child.

We all should understand that child has to improve his knowledge rather than the retention ability.

What is the purpose of this Blog

Yes, I have started this blog with a definite purpose.

I am one of those technical graduates in Electronics and Communication engineering, who eventhough graduated with high marks, did not have the fundamental concepts strong. It is partly my fault as well as partly my instructors who failed to appreciate me of the beauty of the subject , what each subject is meant for, why one should study that subject etc.

The end result was that quite often in my job, when I have to implement something, I sure need to have a base knowledge about it. When I studied it in a different way the fundamentals seems to be quite easy.

Many a times I have wondered what it would be like if the professionals get their fundamentals right during their college days itself. It will make their life simpler as well as prompt them to pursue higher things in their domain when in a job.

A lot of people in India, after their graducation in any technical discipline migrate to the software industry where their work includes testing using tools, editing programmes, creating multimedia tools etc. I am not saying that these are boring jobs. But by doing this for ages his basic qualification becomes quiet useless. It will be always good if we have strong base on what we have learned.

But unfortunately while on their job they won't have much time in finding out how to get their fundamentals right.

So here I will post about the subjects which I have come across, but only its fundamentals so that the time which I have spent for learning it can be saved by some one else reading these posts.

Afterall knowledge which is not shared is meaningless :)

Knowledge Vs Education - A random thought

Can we say that any person who got Educated has gained sufficient Knowledge?
No.
But the questions itself has lots of things to be clearly explained. To anwser the above question we need to find an answer to the following question.

What is the "sufficient knowledge" expected as part of getting educated in a particular subject?
When the experts create a syllabus for a particular course, they have to keep in mind the amount of knowledge expected to be built inside the end user or Student.

Is this enough to ensure that the student will gain intended knowledge after doing the course?
The answer is NO.

For this to happen the following things need to happen during curriculum development.

1. The curriculum/syllabus should be built upon from fundamentals
2. The curriculum/syllabus should contain simple practical application related to the subject
3. The syllabus should contain some excersizes which should make the student think more on the subject.

Most of the time we find that, the course material / syllabus satisfies all the above criteria.

The reason being the fact that most of the time tried and tested educational experts will be behind the making of any syllabus.

Even with this kind of syllabus, we could find a lot of students who must not have gained knowledge as expected by the syllabus maker. Why?

The answer is teaching methodology.

Of late the teaching methods are becoming less effective. It has become completely person dependent.

Professionals end up in teaching at technical institutes since they could not get into any of those lucrative IT jobs. Hence they may not have the passion in them to make students learn

Colleges are focussing on good results in exam rather than the overall knowledge development of student.

The result is that terms like integration and differentiation is not clear to students who have completed graduate courses in Mathematics, even though they have used these for solving a number of problems.

Education industry has grown up as a business establishment where the primary aim is business profit, resulting in value degradation of the educational system.

Who can change all these?

Ofcourse in a democratic set up like that in India where everyone is a leader and where everyone's thinking is self centred and where everyone is thinking about his benefit in doing this, itz difficult to change.

The end result : We will have a bunch of highly qualified professionals who are good for nothing with the exception of a minority who has a self made ability to understand and learn things effectively