Calculus Introduction.

Welcome, This is the first post this blog.
The reason behind this blog is to go deep in the facts and theories which we are using now without knowing the real importance of them. We just follow them because we just want to get marks.
I want you to understand the importance of work of great scientists.
My First topic is CALCULUS.

I wrote this topic in 3 parts, this blog include only 1st part. Here I am going to discuss the basic idea about calculus and just a little introduction. The reason behind 3 parts is that the knowledge is so much and I want you absorb as much as you can so I divided this in three following parts:

1. Calculus Introduction.
2. Differentiation Calculus.
3. Integral Calculus.

NOTE: My aim is not to teach calculus rather my aim is to tell you origin, history, cause of creating, application in daily life and application in pro life.

Introduction :

Well mathematics has various branches, the major branches are-
Algebra
Geometry
Arithmetic
Probability and statistics
Trigonometry
Calculus
etc

Each branch deals with a specific study of mathematics.

Algebra - is the study of operations and their application to solving equations.
Geometry - is the study of shapes.
Arithmetic- is the study of quantity.
Probability - is the measure of the possibility that an event will occur
Trigonometry - studies relationships involving lengths and angles of triangles

Calculus : is the study of change

From other branches of math we find area, volume, angle, possibilities or a specific quantity etc.
In Calculus we also find area, volume, mass etc but with how fast area is changing, how mass is increasing, how velocity is increasing with gravity etc.

In this world everything is changing so fast we have no idea what change can come in our surroundings 1 sec or in 1 day, hour, year, decades.
Hence we use calculus to measure the change.

We can use Calculus as another way like -  what change should I made time by time to achieve a specific target.

Let take some examples where calculus play a very important role:

Example - The logarithmic Spiral.



The logarithmic spiral of the Nautilus shell is a classical image used to depict the growth and change related to calculus

Similarly -Centre of gravity of  Rocket




When a rocket launch and it move away from earth then it leave behind hot gases(which are earlier stored into it) , the mass of Rocket is continuously changing (decreasing) at every fraction of second.
Hence the Centre of Gravity is also changing  every fraction. So it is very very important to fix the point of Centre of Gravity.
Hence Calculus help us first to find that how Centre of Gravity is changing and then change we have to made to fix Centre of Gravity.

(Notice the word I use "Change".)

A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. This can create a realistic environment for movies or video games.

Similarly Calculus is used in advanced programming in Matlab Software to create a algorithm.
The artificial intelligence can be made from calculus.

Hence Calculus has a very major importance in our personal and pro life.




Now some Questions :

1. Why calculus is created?
2. From Where it came ?
3. Who created this?
4. Why we differentiate something?
5. Why we integrate something ?
6. Where should I need to use differentiation and integration ?
7. Why rules Differentiation is consider opposite of integration?
8. Why these rules of differentiation are the rules of differentiation?
9. Why these rules of integration are the rules of integration?




You will every answer of your question in this blog.

History & Origin :

It's a very interesting story and you gonna love it.

The story begin with 16th - 17th century. Before 17th century it was the believe of people that earth is the centre of universe and Sun, planets and whole universe orbit around earth









But in 1543 Nicolaus Copernicus gives the idea that earth is not the centre of universe rather it's one of  9 planets which revolve around sun.
But it was really hard to proof this new theory.

















Later in 1610 Galileo support theory of Nicolaus and made some observations with his own invented small telescope. But still there was no strong proof of this theory.

Galileo faced many problems due to his support of this theory because it was against the church.














In mean time (1609-1619) a German mathematician, astronomer Johannes Kepler gives his 3 famous laws of planetary motion l.e how planets moves around sun

    

.

One of his laws is - The shape of orbit of a planet is ellipse not circular. 
Johannes Kepler died at 1630 and later Galileo died in 1642 but they leave behind the idea of new this solar system called "Heliocentrism" means Sun is centre.

So after that people just start believing about this theory of "Heliocentrism" l.e Sun is in centre.




At the same year when Galileo died, Sir Isaac Newton was born l.e on 25 Dec 1642.  The one of greatest scientist ever. And the one of the founder of Calculus.



All scientist before Newton make him so curious to study about nature. Newton was on the shoulders of  Johannes Kepler and Galileo.
The story of Newton's life itself interesting. At the time of his birth his father was already dead and Newton's mother take care of him son for three years, then she leave him to his relatives and returns after 8 years with her new Husband and children.

This make Newton so upset. Newton love to live alone, he was so curious that how things work, he use to work 7 day in week and 18 hours in a day in his own room.
His was one without family, friends's love who can encourage him.


Now we all know what contribution Newton made in Science and mathematics. But we are here to discuss the Solar system and Calculus.
By the time Newton was young, it was widely accepted that Sun is the centre of solar system and earth and all planets orbit the sun. But now  new questions came  -

 "Why planets moves?"

"What held them in their orbit?"

Many theories came to solve this mystery one of them was Rene Descartes's theory that universe is a giant machine like a clock.

But Newton denied to accept this theory. At that time Newton was studying in Trinity College, Cambridge but in 1665-1666 due to plague disease around the city, the university closed for sometime to avoid infection and Newton returned his home for 18 months.

In these 18 months he give his all time to his work and experiment and finding the answer that how and why planets moves and about the force which held them in orbit.

One sunny and beautiful day he was sitting in front of his house beneath a apple tree with his books and notes.

Newton's House

He was thinking and making notes on planetary force (which keep planets in orbit) and a apple falls down and the new legend born.
He make him think why apple comes down? What attracting that apple ? Why apple don't go up ?
Why it return back when it was thrown upside?

And he realised that - The force which cause apple to fall and keeping the planets in orbit is same.

And then his discovered the Law of Gravitational and Laws of Motion.

But this adventure of Gravity start form Galileo's time when he performed his famous experiment dropping balls from the Tower of Pisa.




Galileo showed that gravity accelerates all objects at the same rate.
Means if you drop two balls one is of light weight like 2 kg and other is of heavy weight of 20 kg then both touch the ground at same time.
Galileo opened the door for Newton, Galileo's work set the stage for the formulation of Newton's theory of gravity.

So Newton start working on this gravity concept, but the problem of gravity can't be solve by simple math because the gravity is a changing phenomenon, when you drop object form a height then its velocity increase 9.8m/s every second.
Hence Newton need a simple mathematical way to prove his views about gravity.
Newton was working on Tangents since 1663 and further he in 1666 he started work on changing phenomenal kind of math l.e Calculus.
In 1671 he created Calculus and around 1679 he formulated his famous formula of Force of attraction l.e Gravitation.



Newton mainly works on finding tangents l.e Differential calculus. Even he does not call it calculus he call it "Fluxions"
And he wrote a book on Differential calculus l.e Fluxion on 1671 and the strange thing is that this book is published after 55 year on 1736.


Even my friend he not yet published his work on Laws of motion and Gravitational law until a good friend came in his life who spread Newton's work around the word.

Who was this good friend??
Well I will tell you but first see how that good friend came in contact with Newton.

Newton got fame earlier due to his work on optics, spectrum of light and his own invention Newtonian telescope.
This telescope contains prism and a reflecting mirror which was far better,cheap and simple in design.



Ok now we came so long. Let see the events again:


Before 16th - Earth centred solar system was considered.
1543 -  Nicolaus Copernicus gives the idea that earth is not the centre of universe.
1610 -  Galileo support theory of Nicolaus
1609-1619- Johannes Kepler, gives his 3 famous laws of planetary motion.
1630- Johannes Kepler died.
1642-Galileo died.
After 1642 -People believing that this theory of sun centreed solar system.
1642 - Newton born
1661 - Newton joined college.
1665 - Due to plague, he returned home
1665 - He observed the Gravity and start working on it.
1666 - Newton start working on Calculus.
1671 - Newton created calculus.(not same as today, improvements made time by time.)
1679 - He mathematically calculated the proof of Gravity.


Ok as I said Newton not yet published his all work. Now come to his good friend.

There was a person who was seeking answer about -

Why planets moves in Elliptical orbit ? (as Kepler said in his 3 theories)
What force held the planet in orbit?
Why planets rotates ?

That person want mathematical proof of these facts and there was no one who can solve this complex problem.

That person and the good friend of Newton was Edmond Halley




Halley was one of the member on Royal Society which job is to publish the good work of scientists.

As I said he was seeking for the answer about Gravity and motion of planets.
In August 1684, he went to Cambridge to discuss this with Isaac Newton and Newton simply said he had solved the problem and he will send a copy of his work to him.




Halley was shocked to see his work recognised the importance of the work and returned to Cambridge to arrange its publication with Newton.
He published Newton's book "Philosophiæ Naturalis Principia Mathematica" by his own money.

First part published 5 July 1687
2nd and 3rd parts were published on in 1713 and 1726.


But Newton was not alone to work on Calculus. There was one more scientist who equally and independently contributed in Calculus. That person was Gottfried Wilhelm Leibniz.




Leibniz was born in 1646.
Leibniz is credited, along with Sir Isaac Newton, with the discovery of calculus.
He mainly worked on finding areas under the curves. Infact he gives the Integration method and the notation of integration sign    ∫   representing an elongated S which means "summa" l.e sum.





Now
Why it was important to find area under the curve?
What are previous methods which were used to find area under the curve?
why Integral calculus is only use to find area?

I will give the answers in 3rd post of this blog because answer are very sophisticated.

On November 11, 1675, when he used integral calculus for the first time to find the area under the graph of a function y = ƒ(x). It was really breakthrough in the field of Science and Mathematics.

He did not publish anything about his calculus until 1684.

Many scientists came after Newton and Leibniz who improve the calculus time by time we not gonna talk about them now but we gonna talk about the basic idea why calculus is created.

Differential Calculus  :

My next post of this blog will explain deeply about differential calculus. I created separate post because this topic is very important, alot of stuff and very sensitive you must need a fresh and ready mind to absorb that knowledge.But Right here I wanna tell you the basic idea behind Differentiation.

So why we use Differentiation?? What will got form that ?

Differentiation mainly deals with to find tangents and slope.Now one must ask "WHAT'S TANGENT & SLOPE??"

Well tangent is just a line but a very important and a very specific line which tells us the gradient of a line or curve.

& Slope is just a number of the tangent line will tells us the magnitude of tangent line, it is usually denoted by m.

Ok lets visualizes this, Take a simple graph:

y=3x  

y=3x  ( a line equation -> y=mx+c)

HERE SLOPE = m = 3;

Ok what's the meaning of this slope now ?

Slope means height(y) changing with length(x).  Here if slope is 3 hence the changing factor remains 3 for entire length.

For 8 steps forward person will climb 24 steps up.
For 10 step forward person will climb 30 steps up.







Ok if a person moves on x-axis then its call   RUN. And if a person will move in y axis then it wall be called  RISE.
And slope is always  Run/Rise.


Here the man climb 24 steps up and 8 steps forward....hence slope = 3.

THI S SLOPE ALWAYS REMAIN CONSTANT UP TO INFINITE...and quite easy to find.

OK now what happen if we increase slope ??

Let take two examples :

y= 4x     &     y= 5x




OK now y = 4x    l.e slope = 4






Person has to climb 24 in 6 steps.


Now see y = 5x    l.e    slope =5 ;



Person has to climb 30 steps up in 6 steps forward ( he has to work more harder than previous)


Now let see slopes increasing for 1 to 6  in y = nx







As you can see with increase of slope Rise (Y) increases and Run (X) decreases.

Ok so what can you do from this result?

Let x represents TIME,
y represents    PRODUCTION QUANTITY.

So now your aim is to reduce "x" l.e time and Increase in "y" l.e production
Or more shortly your aim is to increase slope by your efforts.


Ok Now we just seen what are tangents and slopes but we not yet discussed about calculus.


OK as you see it was quite easy to calculate -
1. Slope (by Rise/Run)
2. Area Under line (geometrically)
3.Length of Line

As shown below:






But what about curves like below:




What's its slope ?
What's its length ?
Whats area under this curve ?



Well its slop is not fixed as Previous example. Slope is changing at every single point so it's become hard to find answers of above three questions.
But with the help of Calculus we can find all answers.


Let take Y= X^2





by differentiation its slope will be 2x.
Hence for any value x we can find exact value of slope l.e  "m"   and after finding slope we got the idea that how X is changing with Y.

But why we are finding above stuff for curves ??
Well look below:

Everything in this world is pretty much curvy.










Hence is  very important to analyse the curve.
But as I said there was simple math before calculus which can analyse these kind of curves.
Hence calculus is created.

As we know Calculus is kind of math which measure change.

Well this is just an intro...there is alot more we can do by calculus.
In my next post of this blog i will explain Differentiation more deeply and also tell you how differentiation made our life easy.

Integral Calculus  :

We will not go in deep in this topic because integration itself  a  huge and mostly use in engineering.
I have planned to make 3rd post of this blog.

But right now i want you to know that We use differentiation to find tangents on curve.
And we use integration to find the area under that curve.
Ok thats it.

I see you  in next post of Differential calculus soon.
I hope I clear most of your doubts.




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