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?