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 .

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WHAT IS THE NATURE OF NATURE : HOMOGENEITY OF SPACE AND TIME

                                          

                            UNIFORMNESS OF SPACE AND TIME

Two people were playing with concave and convex mirror named as Dev and Divya 

Divya , " Don't you think if mirror world were real the there were no uniformness in that universe ??"

Dev, " Such a interesting topic to think on ??.... But"

Divya , " But....What 

Dev , " What does uniformness of space means" 

Divya , " Let me explain"

Dev, " Yes , Go ahead , Mam!!! ."

UNIFORMNESS..........

As the name tells uniformness means being uniform.
For Example - Sugar mixed in water is uniform.
                         A car moving with same speed have uniform motion.
                        A Road with no pits can be called as uniform

But what is the uniformness in space

........SPACE AND UNIFORMNESS

We are going to talk about uniformness of space.

What does it uniformness of space mean ?

It means that two point in free space have no difference. Any experiment performed at both place will have exactly same results.No point is space is special, so the same basic laws of physics should govern all of space.If two electrons repel each other with force of about 4 units on earth then by keeping their distance of separation same and you take it to andromeda galaxy the repulsion will be same.

Let us elaborate it.

Let us two observers A and B.
Now we are going to find the relation between the co-ordinates systems of two observers.
Let us consider observer A is at rest.Then for him  if an object moved from point (0,0) to (2,0) and then from there to (4,0) . Then in both the cases 

Displacement Δx = 2 metres

According to space homogeneity law

Any other observer in another frame of reference will see equal displacement in both the cases.
I  mean to say that space will follow linearity relation between the two frame of reference.

Then we can state that

Δx = kΔx'

                                                  where k is constant
                                                             and Δx ' displacement in the frame of B

As this is a linear relation between space co-ordinates this is also known as

"Linearity of Space and Time"

Could you figure out what will happen when if space will not homogenous??

Let me help you out.

Then there will not be linear variation between the space co-ordinates of two frames of references.
 Let say they follow a relation as

kΔx' = Δx2

Then as object moves from (0,0) to (2,0) in frame A . Then it will move

from (0,0) to (4/k , 0) in other frame B.

While when it will move from there to (4,0) in frame A. Then it will move from  (4/k , 0) to (12/k,0)
in frame of B.


This implies space is function of its position in frame B.This not only changes the displacement but will also change the entire physics.

For Examples -

# 1 meter stick in A frame will change it's length in frame B w.r.t. to its position in A's frame.

It will become 3/k m when put between (2,0) and  (1,0)
and will become 5/k m when put between (3,0) and (2,0).

# Electronic repulsion will also vary in B's frame when entire electronic system will  shift.

# Energy conservation law will be violated in B's frame

Same homogeneity law goes for the time too.

Could you figure out what will happen if time was not homogenous ???



THANK GOD !!! WE ARE NOT LEAVING IN AN MIRROR UNIVERSE.

OTHERWISE WE WILL BECOME FATTER AS MOVE IN PARTICULAR DIRECTION .

THANK YOU GOD !! SPACE AND TIME ARE HOMOGENOUS AND ISOTROPIC{same in all direction}......

MATHEMATICAL PHYSICS : HOW LARGE INFINITY EXACTLY IS !!!!!

           

                                         INFINITY

..................A STORY OF BEGINNING OF RELATIN OF HUMANS AND NUMBERS

Around 4,000 years ago , A shepherd was counting his sheeps with the help of stones as there was absence of language to count.He had 20 stones for his 20 sheeps. He used to take all his sheeps for a walk by counting it with the help stones.He used to put out one stone as one sheep comes out from the house and other for other sheep and so on.......

MISCONCEPTIONS ABOUT INFINITY......................

#1. There is common misconception about infinity that is 

                                     1/0 = ∞

Which is not correct as 1/0 is an undefined quantity.We cannot come a conclusion with this expression.

So, what is the truth ?

          1/0⁺   = ∞
0+ means a no. just greater than 0.

Let us elaborate it

0+ could be equal to 0.0000000000000000000000000000000000000000000000000000000............. write zeroes until you die and ask your son to do the same until he dies and ask him to ask his son to do the same and let the cycle continue after about (10000000000000000000000000000000000000000000000000000000....... zeros where you can think off) generations  , you take a rebirth and you write zeros all your life and before your death you put 1. The no. you will get that will be 0+ .

And when you will reciprocate that no. you get a no. that is known as infinity.

SIMPLY IT IS ENDLESS...............................

#2. There again a common misconception about infinity that is

Its a no.

No , Infinity is not a no. Its a concept to create such a large no. that you cannot think off. As we saw in 1st case after following the steps you get a no. that much large no. which you cannot think of.

If think that you can then let me break you myth.

Do you know about avogadro no. which has value of 6.023 * 10²³ ?
Let me explain this thing to you

If you get this much money and you starts spending 1 billion dollar per second then and live for let say 150 year even after that you will left with about 99.999999% of money left in you hand.

Getting that point just consider 10¹²⁰
and think how much that no. would be. Just think about that no. which you cannot think off.

#3. There is another common misconception about Infinity that is
When there is a series which has infinite terms that's sum will tend to infinity
About which We had discussed about it in our previous blog.

            LOOKING FOR INFINITY...........

If I ask that " Have you ever seen INFINITY ?" Most probably your answer is NO.

But I can bet that you have ....

In your barber's shop........

There two plane parallel mirror placed in front of each other  which have the tendency to form infinite no. of images. You could find infinity there.

.............GETTING RELATIVELY INFINITY

Now we know how the real concept of infinity and how infinity looks like.Its time to tell you that not always infinity is an absolute term. It may be relative.

Like in the case of electromagnetism we have formula for infinitely long solenoid's magnetic field as

          

                                                                   B=𝜇 n i

                                                             where 𝜇 is constant

                                                                         n is number of turns per unit length

                                                                         and i is the current in the wire

As we know that it is not possible to create infinitely long solenoid . So, why did we give birth to this equation ??

A simple answer to that we could it by applying some conditions

Let length of solenoid tube be L and R be the radius of solenoid .
Condition is L>>>>>>>>>>>>>>>R.
Let me tell you that it is very common condition.

.................UNDERSTANDING INFINITY IN A BETTER WAY

Let take one more step to understand infinity.

We will do that by connecting to our first case.

Let us take an example.

There were two children named as Devendra and Divyanshi  and a game was going on between them.According to rules of the game the child who would shout out the bigger no. would win.As they are children so they don't know the no.s bigger than 100.

   

Devendra said , " I want the game to be a little bit longer. So, I would start with 1."

Divyanshi said, " Me too. So I will go with 7"

Devendra said, " You just called out my lucky no. I will call out your 9."

Divyanshi said , " I am forgiving you the last time. I will not show the mercy next time . Thats why I will go with 25."

Devendra said , " I will not show mercy even this time 100. GAME FINISHED!!!!"

WHY DID I TAKE THIS EXAMPLE ??

I took this example to make connected with the very first story and starting of relations of humans and no.s.
As I told you in the first story , In ancient times, In many of the civilizations there was no word for a no. greater than 3.So, how they used to count ?? They used the method of putting marks on the walls for each new element greater than 3.Ancient civilizations are the basis of modern number theory which has this concept of infinity.
So, in ancient civilizations counting infinite means putting infinite marks.It is a similar case to putting zeros for generations as we did earlier.

Let us consider a case that there were two people trying to count the infinity. So, putting marks on walls.The speciality is that wall is infinite and both of them have infinite life.First one is doing that with the rate 2 marks per second and Other one is a little bit lazy and doing that with rate 1 mark per second. Will they have same no. of marks in the end ???

Let us make a final approach to infinity.....

FINAL APPROACH TO INFINITY

Let me start this section by stating that there are as many as even no. as many whole no. which means even no.s and whole no.s are equal in their no.s .

"How is this possible ???"

If this statement is true let us try to reach up to it.

Then we have understand the meaning of having exactly same no. 

Meaning of having same no.of two different quantities is that we can overlap the them.

For Example we have two hands and each hand have same no. of fingers i.e. 5 we can make them overlap.

 

Let us try to use this logic to prove the above statement

1 *2 = 2

2*2 = 4

3*2 = 6

4*2 = 8

5*2 = 10

.   .       .

.   .      .

.   .      .

.   .     . 

So , we have an even no. for every no. Hence we proved it.

Now , you will state that but between these no.s

1, 2,3,4,5,6,7,8........

we have 2,4,6,8....

I know you are doing correct but this procedure makes us to count half way.

To avoid confusion Don't go for counting when we have an even no. for every whole no.

Let say you want to go for it.

Then , For Example-

You count as 

1,2,3,4..... then we have four even no. too.

2 corresponding to 1

4 corresponding to 2

6 corresponding to 3

8 corresponding to 4

But as you counting stops to 4 you get only 2 no.s instead of 4.

You left 6 and 8..........

When you are doing this you are actually applying the wrong rules of overlapping....... As putting all the fingers on a single finger of other hand.

So, how this statement was related to UNDERSTANDING INFINITY.........

Actually it tells us about a beautiful and amazing nature of infinity. This explain clearly why infinity is not a no. and is a concept!!!!!

So, we can say even if a person is putting marks with the rate 1000 marks per second and other one is writing it with 1 mark per second after infinte time they will have same no. of marks on that wall.

Similarly we can say that there are as many as Integers as many as whole no.s

But are same no. of prime as no. of Integers.??
YES

PROOF
Suppose there are only n prime no.s. as follows

P1, P2 ,P3, P4................Pn

we want to construct a new no.

N = P1, P2 ,P3, P4................Pn +1

Now this no. is not divisible by any these prime no.s as remainder will be 1.
So, we produce a new prime no. which is greater than all the previous ones. So, prime no. are also infinite.

Now let us go more deep into it.

Let us talk about decimals and their comparison with whole no.s

Here , we are going to compare two infinity sets.

Now we cannot state that there a decimal for every whole no. As there so many so many decimals. We cannot even list all the decimals.But we can do that for some in a way known as Contour's Method.
George Contour was a great mathematician.

1/1 , 1/2 ,1/3 , 1/4 , 1/5 , 1/6 ,1/7, 1/8 ................................
2/1, 2/2, 2/3 , 2/4 , 2/5 , 2/6, 2/7 ,2/8 ,2/9..........................

Like this we can make a list of all the Fractions
There is no way to make the list of irrational no.s

From above we can definately state that Fractions also forms an infinite set.

But we can say that for Fractions there are as many as whole no. as many as whole no.s
Because of following logic

But we cannot do that with the irrational no.s. There is no way to list the irrational no.s.
 Because you will always be left infinitely many decimals.

For example -

You decide to put        1.112121211244564678899532.......... in your least
But you forgot to put   1.112121211244564678599532..........

If you that too , I will always to create a no. that is not in your list.



This is called UNCOUNTABLE INFINITY
In case of rational no. we say it COUNTABLE INFINITY

This Implies following

Which implies that THERE ARE DIFFERENT SIZES OF INFINITY....WHICH VERY DIFFERENT ASPECT TO THINK IN......

But why discussed it all ??

Because I want you take to the topic CONTINUUM HYPOTHESIS.

CONTINUUM HYPOTHESIS

Roughly,

It states that there is no set which has no. of elements between whole no.s and decimals....

Let us be more formal
"There is no set which has cardinality between integers and real no.s
More simply real no.s is the set which has possible cardinality just greater than integers ."

Cardinality simply tells the no. of elements in the sets.

Contours' Theorem -

"Let A be a set and P(A) be its power set then cardinality of A is strictly dominated by cardinality of P(A) ."

Power set is just all the possible combination of elements in A

Then if cardinality of A is 4
Then cardinality of P(A) is 2^4  i.e. 16.

On the basis of above theorem and using the fact that real no.s are power sets Georg Contour hypothetically thought that there no infinte set which has cardinality between integers and real no.s.

As I stated that it is a hypothesis of a great mathematician GEORG CONTOUR.

Till Now we neither have a proof to prove this hypothesis correct nor have a proof to prove it wrong.....



  THAT"S WHY DAVID HILBERT TOOK THIS AS ONE OF THE UNANSWERED QUESTION OF MATHEMATICS........

FROM STARTING OF ANCIENT CIVILIZATIONS WE ACHIEVED A LOT.
WE FOUND A LANGUAGE TO COUNT,
WE ADVANCED A LOT IN TECHNOLOGY ,
 WE HAD CAUGHT MOON AND ONE OF OUR DESIGNED MACHINE IS ROAMING ON MARS.
BUT WE STILL ARE UNANSWERABLE TO A LOT OF QUESTIONS, ONE THOSE IS HOW LARGE INFINITY EXACTLY IS!!!

GOD ALREADY KNOWS IT VERY WELL , HUMANITY HOPES TO  GET A MAN WHO WILL KNOW INFINITY!!!!!!

BUT STILL THIS TOPIC IS NOT GOING TO END BECAUSE IT IS INFINITY.........

MATHEMATICAL PHYSICS : MAKING SERIES

                                                 INFINITE SERIES -1

Series is basically collection of terms following a particular pattern.
Infinite series are those series which infinite no. of terms.

Didn't get that ?

Let's us try to it a little bit more interesting......

Consider a thought experiment , you have decided that you will jog continuously till you lose your belly fat , but with a particular pattern . According to that pattern you will jog 1KM in first hour then 1/2 KM in second hour, 1/4KM in third hour and so on.....

               Distance you travelled to lose your belly fat = 1+ 1/2 + 1/4 + 1/8 +1/16+......
       
 Above is an example of infinite series . Total distance you travelled can be summarised mathematically by
                                     

                                                Sn =   ∑_n=0(1/2ⁿ)
                                     
                                                            where n tends to infinity
 
Consider another thought experiment , you are on a beach and you decided that you will take out all the sand from it by using a glass which has capacity of 1000 Ltrs ,but with a particular pattern.
According to that pattern you will take 1000L in first time then 1000/2 Lin second time, 1000/3L in third time  , 1000/4 L in forth time and so on.....

                         Volume you took out = 1000 ( 1 + 1/2 + 1/3 + 1/4 +......)

 Above is another example of infinite series. Total volume of sand you took out can be summarised  mathematically by

                            Sn =  1000∑ _n=1 (1/n)
                                       
                                               where n tends to infinity

        So, we are clear with the concepts of infinite series.

Now there is a basic question that can be arisen in this case i.e.

What is the sum of such series?

The very obvious answer that comes from students' side is infinity which roughly means cannot be counted i.e. such a large no. that counted
The is half correct as it may not infinite. It depends on the type of series.

Again consider a thought experiment, you have a square and you have decided to make cuts on the square such that every succeeding piece has half of the area of preceding one.
As shown

                 

In above if add all the cut outs after making infinite terms you will not get more than area of original square.

       Here series goes like this

                           A + A/2 +A/4 +A/8 +A /16.........
It is geometric progression , which means succeeding and preceding term has constant ratio.
A G.P. can be defined as a progression of which terms are in a constant ratio.It could be increasing or decreasing.
Example - 3,9,27,81....so on. is an example of increasing. Here const. ratio is 3
                 -3,-9,-27......so on is an example of decreasing. Here const. ratio is 3
A G.P. can be given by
           
                              a , ar , ar²,ar³......
where r is the constant ratio .
Sn be the sum of n terms. There is a particular trick to get this sum. I am doing this r <1
                     sn = a + ar + ar^2  +  ar³......ar^n-1
                  -  rsn = -ar - ar²  -........... ar^n
             

                 (1-r) Sn = a(1-rⁿ)

                   Similarly it could be re-imagined by considering r> 1 keeping in mind that sum of                              positive must be positive.
                After doing that we found following result

                             For r >1 as n tends to infinity
                             Sn also tends to infinity
                              For  r<1 as n tends to infinity


                                                  Sn = a / (1-r)                          rⁿ  will tends to 0       

                                 where a is the first term

      From this we could state you could lose your belly fat after jogging just 2 Kms.
 
BE THANKFUL TO GOD THAT YOU NEEDNOT TO RUN INFINITE KMS!!!!!

Let's come back to our problem. Till Now we discovered that there are two type of series


1. Those which converges at a particular point CONVERGING SERIES
2. Those which diverges DIVERGING SERIES.

Now our problem is to determine whether the series is a converging one or diverging one.

We have some tests to determine that :
1.Comparison Test
2.Ratio Test
3.Integral Test
4.Special Comparison Test

Before doing these tests we should do PRELIMINARY TEST
In this test we convert the in terms of series as a function of 'n'. Then we check what happens when n tends to infinity.
Then if they also tends to infinity then series is a diverging one, if NOT GO FOR SOME OTHER TEST.

Example -

         Series like 2, 4, 8, 16,......

              'n'th term = Tn = 2ⁿ
As n tends to infinite Tn also tends to Infinity . So it is a diverging series.

Let us consider another series.

1 +1/2 + 1/3 +1/4...............

Tn = 1/n

As n tends to infinity Tn tends to 0.
This means that we need to go for other test.


REASON FOR ABOVE CONCLUSIONS
When Tn tends to infinity we said that the series is diverging one that seems very obvious . But problem arises when we said that when Tn doesn't tends to infinity why can't we conclude anything?
This is because in such series always there is addition of terms which have enough magnitude to increase total sum value....

Let's take an example:

Consider a series as following
1 + 0.0001 +0.00000001 + 0.000000000001 .......so on
Even after adding nearly 10,000
 It just become equal to nearly 1.0001.......
        Tn tends to 0

But In following series

1+1 +1 +1+1....... so on
After adding just 20 terms we get 20 .
            Tn tends to 1.



 So, we cannot use the preliminary test to test all type of  series. It is limited.

1. Comparison Test
It is the most important and the basic test. It gives birth to other tests.
Generally experienced mathematicians use it to determine whether series is converging or diverging.
In this test we use known series to test its nature.

For Example - We know that you need not jog infinite kilometers to lose your belly
                        which imples
                                1 + 1/2 + 1/4 ..... is a converging series.
                 
          Any series whose terms tends to a smaller value than the above one will be converging series.

            Let us elaborate this point .......

         Let us take series
       
                 
             1 + 1/3 + 1/9 + 1/27...........

             which could be summarised as

                               Sn =   ∑_n=0(1/3ⁿ)

    We can easily tell that for same value of n
     second series will have smaller value .
    As we know that the first series converges we could easily tell that second series will also                   converge because for any n second series will have less value than first one.

        As the formula of sum of GP states that it has to converge at 3/2 , which proves our point.


To test whether a series is diverging or not we have to check whether the given series tends to larger value than any diverging series.

Let us elaborate this point.......

For Example -  We know that
                           
                               1, 2, 4, 8,......... is a diverging series from the preliminary test.
               
                   Let us take a series

                                     1,3,9,27,......
                                       
                                Sn =   ∑_n=0  3ⁿ
         
  We could easily compare the two series and tell that as n increases second series will become larger than the first one.
So, the second one is also diverging.
It is so because the sum of first series tends to infinity and second series has to a bigger value than that.


2.Ratio Test
It is just the modification of comparison test .

It states that first find the ratio of T_n+1 and T_n.  
Consider that ratio as r
if r<1 for n tending to infinity then series is converging
if r>1 for n tending to infinity then series is diverging
if r=1 for n tending to infinity then go for another test.

For Example- Let us take a series

                   1/1! + 1/2! +1/3! +1/4!./........
{ 1!= 1, 2! = 1*2 , 3!= 1*2*3 , 4! = 1*2*3*4......so on}


Which can be summarised as

                   Sn =   ∑_n=11/n!
Now consider a situation that a energy source generator is generating energy as above series as energy per second and we want to calculate whether the total energy is infinite or a finite value. So first we need to find whether this series is converging or diverging.

Let us do it by ratio test......
       

      r = Tn+1 / Tn

            = 1/(n+1)! ⨸ 1/n!
         
                 = n! / (n+1)!
                   

                        =  n(n-1)(n-2)(n-3)................ / (n+1)n(n-1)(n-2)............

                          = 1/n+1

            As n will tend to infinity r will tend to 0
 This implies energy has a finite value.

Now let us check a previous pending case of sand at the beach.

 i.e. the series which has summary as

                                  Sn =  1000∑ _n=1 (1/n)

                r = (1/n+1 ⨸ 1/n)
                     = n/n+1
                         
                          = 1 / ( 1 + (1/n))

as n tends to infinity r tends to 1.
SORRY !!!! Its the third time we are failing to determine the type of series.

PROOF OF RATIO TEST

After observing the above procedure we could state that we are finding limits of ratio Tn+1 and Tn.


For those who dont know what limit actually is
Its a mathematical tool about which we will talk later in detail. But , For Now Its just a method with we were able to find the no. to which r is tending.It has exactly the same procedure that we to simplify but with a f**king type of symbol which has no meaning.

Now as we finding the limit of r an n tends to infinity let us denote it by L .

3.Integral Test