MATHEMATICAL PHYSICS : STARTING CALCULUS

      STARTING CALCULUS........

CALCULUS : DISCOVERY OR INVENTION 

Calculus is a tool to study the mathematics of change in nature.Newton and Leibnitz led the foundations of calculus. But the question is .....

Was it discovered or invented ??

It was invented to understand the real nature of Nature .Newton and Leibnitz did it separately . Newton did it for physics and Leibnitz did it to understand the variations in functions.

WHAT IS THE NEED  OF CALCULUS??

It is going to start with a story of a husband and wife , Dev and Divya .

Dev is driving a car and Divya is riding.

Divya suddenly asks ," What is our speed ??"

Dev  says , " But speedometer is not working .How can I tell you ??"

But Divya like any other wife is stubborn . So, she needs an answer.

Divya says, " I need an answer . Not an excuse."

Dev , " Ok. We travelled about 30 k.m. in last hour . So our speed is 30 k.m. / hr."

Divya is a mathematics student . So , she knows that Dev is talking about average speed over an hour.

Divya says , " Dev don't fool me. It's an average .I want to know the current speed."

Dev ," Ok !! So , last minute we travelled 1km. So , our current speed is 1km / min."

Divya , " That's an average of last minute. 

I don't want the average of even 1 sec. Let's be specific find out the speed when our car's middle point was in the front of that pole."

Dev, " So, our car travelled 0km and time taken to travel that distance is 0s. So our speed is 0/0 m/s. But what is 0/0 ?" 

This problem gave rise to the solution CALCULUS.

But how can we solve this problem ?

Let's say x is distance travelled by car in t time . Then average speed is
     
                      x / t

But as t will become smaller, average will become more accurate. For example average over minute is more accurate than hour.

Let's say car travelled 0.0009 m in 0.0001 sec. Then it's speed is average of 0.0001 s i.e.
9m/s . But average over 0.0000001 s will be even more accurate than previous one. To get instantaneous speed make t's approach towards 0. This gives birth to Limits and Derivatives

             Instantaneous speed in language of limits can be written as

It simply means that make t smaller as much as you can think.

We will extend this topic more in next post about it .