Mathematics is about numbers, physics is all about mathematical functions..

Theory of everything

Theory of everything is the ultimate theory of physics , that should describe all physical aspects and occurrences in a consistent manner. Such theory , if it is found, needs to unify all fundamental forces of nature. String theory is the best candidate for theory of everything. It is a theoretical framework that have merged quantum mechanics with theory of relativity. According to string theory universe has nine space dimensions and one time dimension. All the dimensions other than three are curled up and tiny so that we can not directly perceive them. Besides this crucial feature, the principles of string theory say that what appears to be elementary particles are tiny vibrating pieces of strings. Somehow if we can work out resonant or vibrational pattern of the piece of string we will be able to explain the observed properties of the elementary particles. The conclusion is that the universe is a cosmic symphony of strings resonating through ten dimensional hyperspace. What are we ? We are nothing but the melodies. We are nothing but cosmic music played on the same cosmic strings or membranes.
General theory of relativity theory of quanta

General theory of relativity + theory of quanta = Everything

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General advice

Readers from all over the world , have you a feeling that you are bad at mathematics? Do not fear mathematics. Once I was also a victim. You can now conquer mathematics easily. The secret is behind logic. Logical ideas alone can define mathematics. I will state russell's paradox first. If you understand this you can safely assume that you have a good mathematical sense .
Suppose R is a set of all sets that are not members of themselves. Now is this set R is a member of itself or not? If it is a member of itself then it is one of the sets which are not members of themseves. So it is not member of itself. And if this set R is not member of itself then by definition it is a member of itself. So in either way we come to a contradiction!

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Mathematics

Mathematics is the language of nature. It is hard to define mathematics . In general it is the study of change, numbers and shapes. Mathematics consists of various branches like arithmetic, geometry, topology, calculus and many more. Science can not progress without mathematics. This is so obvious fact that needs not be mentioned. Mathematics , in a certain sense , an exact science. We all accept the mathematical theorems and truth because we can prove those. we all accept 2+2 = 4. Mathematics is always flawless. That is what mathematicians thought every time. But Kurt Godel gave a death blow to mathematics by claiming that no mathematical system is complete. Axiomatic systems are necessary incomplete. That means there will always be some statement within the system , whose truth can not be proven using the system. This was very frustrating idea for the mathematical society and no one has yet been able to refute Godel's claim. This is known as the Godel's first incompleteness theorem. This theorem is kind of related to the "liar paradox" of usual language. Suppose there is a sentence " he is lying". Now if the sentence is true then when he says that he is lying , he is indeed lying. That means he is telling the truth. The sentence's truth can not be determined as we come to a contradictory situation. There are other similar paradoxes explained here.
"Mathematics has a better sense than the common sense"
There is hardly any branch of engineering and science where mathematics does not apply. Mathematics is a sense you never have. So nobody is gifted with mathematical ability. You can not be good at it without practising it. But it seems surprising that our mind which is so well organized can not adapt with mathematics everytime. Or may be it is a wrong idea.
"pure mathematics is a poetry of logical ideas"

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The invention of calculus has changed the world. Calculus was invented by Leibniz and Newton at the same time although there is priority dispute about who invented it first. Application of calculus is everywhere. Where there is change there is calculus. Where there is curvature there is calculus. Both the theory of relativity and quantum mechanics were developed using calculus. Everybody knows that the idea of calculus is rooted in the notion of limit. When we speak about something approaching another one or some point approaching another point , the notion of limit appears. There is precise definition of limit. You can think of it as a scheme to manipulate infinite number of small quantities. We know we can sum infinite number of terms in a geometric series, which converges. Same thing happens in case of limit. One quantity converges to another and in the end they become equal. This is the core idea of limit and calculus.
integration
Real numbers are all the numbers that can be traced on number line. It includes both rational and irrational numbers. Real number line contains no gap or it is continuous. Irrational number was defined in order to make the real number line complete. It is the set R. It is extended from - (infinity) to +(infinity) including zero. Real numbers are also called measurable numbers. It includes the set of integers too. Calculus is certainly established using the concepts of real numbers. Cantor first proved that the set of real numbers is non denumerable. It can not not put in one-to-one correspondence with natural numbers(1,2, 3, 4, 5,6 ......). Real numbers has cardinality 2^(ℵ0) , which is called cantorian continuum. Cantor also proposed that there is no set which cardinality lies strictly between that and ℵ(0). This is called continuum hypotheses. These two infinite numbers of continuum hypotheses play exactly the same role as integers 0 and 1. There is no integer in between 0 and 1. Cantor knew that this was the case but he never came up with a proof. Later Kurt Godel showed that this hypotheses is consistent with axiomatic set theory.
Pythagoras law has made a great impact on science and mathematics. This law holds for any right angle triangle in Euclidean geometry. The law says square of the hypotenuse is equal to sum of squares of other two sides. This law not only bears the fact of geometry but also laid the foundation of many scientific theories like theory of relativity and electromagnetism. Pythagoras law also states that there is a 3-tuple of integers which satisfy it. In another words there are infinite number of solutions of Pythagoras law. More specifically , there are infinite solutions for x, y and z in integers which represent the three sides of any right angle triangle. This brings us to Fermet's last theorem. The theorem states that there are no solution for x, y and z in integers for power greater or equal to 3. Proof of Farmet's last theorem took much time to be discovered. This proof involves number theory.
Gauss was perhaps one of the greatest mathematicians of all time. He was first to study the curved surfaces. What we call plane is particular cases of 2-dimensional surfaces or manifolds. In physics he contributed also. We all know Gauss's law of electromagnetism. Gauss was very talented in his early childhood. One day his teachers asked everybody in his class to add all the numbers from 1 to 100. Gauss calculated it more quickly than others within seconds. He just did it by adding two numbers and multiplying it by 50. How ? 50 pairs of numbers have the same summation like 1+99 =100 , 2+ 98 = 100, and so on.. So the total sum is 50*100 = 5000. Gauss also developed many theorems which are still very usefull.
"Mathematics gives you wings"
Roger Penrose is the world's leading mathematical physicist. He collaborated with Hawking and proved that time had a beginning. The proof concerns the postulates of the theory of relativity. He is a mathematician too. He is specialized in recreational mathematics. He makes many complicated mathematical problems seem easy by graphical representation. Mathematics, according to him, has certain platonic existence.

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There are three kinds of worlds :one is physical world, second is mental world and the third is platonic world. Mathematical objects and forms exist in platonic world. Mathematical objects or forms are numbers, triangles, circles, sets and many more. There is no perfect circle in real world. We can not see perfect flat space anywhere in the universe. This kind of pure mathematical or geometric objects exist only in the platonic realm.
Our mental world consists of all our thoughts and abstract percepts. It has no connection to physical reality. Our mental world contains particular or percepts. Universals are certain concepts which can exist without our mind. Mental world does not necessarily make logic and mathematics dependent on it.
Physical world is the materialistic world that we inhabit. It is the whole universe or multiverse-as we may call it.

carton
So what is the most complicated topic in mathematics? Is it topology or geometry or calculus? or is it linear algebra or statistics? It is none. Mathematics builds upon itself. If you want to understand calculus you need to understand limit and functions. If you need to understand topology you need to understand set theory. It is like a chain. If one link is missing the whole system is broken. We do not understand mathematics because of these missing links. Lot of things is dependent on definitions. We can define something when there are notions which have a certain relation to some term , which is itself one of the said notions. Why is physics hard to understand? Physics has lots of notation and symbols which scares us away. You can talk about general relativity. In relativity all boils down to 21 numbers that Riemann curvature tensor assigns at each event in spacetime. In classical electromagnetism at each event in spacetime there are 16 numbers that electromagnetic field tensor assigns. How does these numbers creep in? I guess you have to have some ideas about those theories. Take another example of quantum field theory. This theory treats particles as excitation of the fields. Everything in QFT can be thought as perturbation in an infinite number of sets of infinite number of tiny springs. Particles come from energy that the fields represent. In quantum electrodynamics there are three functions which give three different numbers describing all interaction between matter and radiation. I guesss I have to say that Pythagoras was right all along when he said that numbers rule the universe. If you know only calculus well and have some basic ideas of linear algebra you can understand a lot of advanced theoretical physics theories.

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"If the multiverse theory is correct then there is someone exactly like you (doppleganger) which is reading this sentence in another universe!!"
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Chaos Theory

Some systems are very sensitive to initial conditions. A small change in the initial conditions can create a huge disturbances in the system. Such system is known as chaotic system. For example, a flapping of the butterfly wings in New York can cause a hurricane in London. This is known as butterfly effect. Mathematics of such system is very complicated. To analyze chaotic system chaos theory has been developed. The dynamics governing such chaotic system is also very complex. We can not predict the behaviour of such system as exact initial condition is always unknown.


Miscellaneous pages
Algebra   |  coordinate geometry   |  Topology for dummies   |  Symmetry   |   Bertrand Russell philosophy   |  Sir issac newton   |   Statistical mechanics   |  Albert Einstein  |   Artificial intelligence   |  Leonhard Euler   |  Carl Friedrich Gauss   |  Number theory   |   what is science   |  complex analysis   |  perihelion of mercury  |   string theory explained  |  Calculus and differential equation   |  Medical Science definition   |  Mathematical universe   |  Calculus for dummies   |  Weyl's theory  |   electron and proton
Complex analysis
quantum mechanics and elementary physics |  Mathematics (Advanced)

"The best thing about science is that you can apply it to your life"

Quantum Field Theory

Feynman's sum over histories   |   S-matrix   !   quantum field theory

Theory of relativity

Relativity made simple   |   Special theory of relativity   |   General theory of relativity   |   Tensor calculus  |   Hamiltonian mechanics   |   Field equation  |   Perihelion of mercury|   Geodesic distance

Quantum mechanics

Schrodinger equation   |   Matrix mechanics   |  Dirac equation

P VS NP

P VS NP problem is one the hardest problems in mathematics. If you can solve this problem you will win a million dollar prize from clay institute of mathematics. P here means those problems which can be solved in polynomial time. It is deterministic set of problems. P also means that a problem is solvable by Turing type algorithm. Whereas the NP means indeterministic problem. Given a problem there are altenative choices. And one of those alternative there are more alternatives to choose from. A particular set of steps from those will determine the problem's solution. But a prior one would not know which steps to follow to solve a particular problem. Hence it is NP (non-deterministic).

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heisenberg Real number is the basis of mathematics and theoretical physics. What is real number? A real number is nothing but a class of rational numberss. The class of rational numbers less than 1/2 is a real number. This was the consequence of the theory of real number. The irrationals were found to be some series of rational numbers, which has no rational limit nor infinite limit. And real numbers are the totality of rationals and irrationals. This was the idea behind defining real number as the class of rational numbers. The whole theory deserves another page , which will be made very soon.


hamiltonians jacobi equation

Proof of the pythagoras theorem

Let QR = a, RP = b and PQ = c. Now, draw a square WXYZ of side (b + c). Take points E, F, G, H on sides WX, XY, YZ and ZW respectively such that WE = XF = YG = ZH = b.
pythagoras law
Then, we will get 4 right-angled triangle, hypotenuse of each of them is ‘a’: remaining sides of each of them are band c. Remaining part of the figure is
pythagoras law
square EFGH, each of whose side is a, so area of the square EFGH is a2. Now, we are sure that square WXYZ = square EFGH + 4 ∆ GYF

pythagoras law
That is to say, hypotenuse of right angle triangle squared is equal to the sum of base squared and verticle squared.

Physics of everything


equations of everything physics


equations of physics


equations of physics

equations of physics
Love formula
equations of physics

P VS NP

P VS NP problem is one the toughest problems in mathematics. If one can solve it he will get a million dollar prize from clay mathematics institues. P means that a class of problems is sovable by Turing type algorithm in polynomial time. On the other hand NP is non-deterministic. Given an alternatives there are other alternatives to choose from. A specific set of steps will determine the solution of the problem. But a prior one would not know which alternatives to follow. Hence it is non-deterministic.
equations of physics

Different proof of the irrationality of root 2

root 2 is the diagonal of a square whose side is unity or 1.
proof of irrationality root 2
The issue is to see why the equation (a/b)^2 = 2 does not have any integer solution. Where we take these integers to be positive. We shall use proof by contradiction to prove that no such a and b can exist. We therefore try to suppose, on the contrary, that such an a and b do exist. Multiplying the above equation by b^2 on both sides, we see that it becomes
a^2 = 2b^2
and we clearly conclude that a^2 > b^2 > 0. Now the right-hand side, 2b^2, of the above equation is even, whence a must be even (not odd, since the square of any odd number is odd). Hence a = 2c, for some positive integer c. Substituting 2c for a in the above equation, and squaring it out, we obtain
4c^2 = 2b^2 or b^2 = 2c^2
and we conclude b2 > c2 > 0. Now note that the corresponding integers are smaller than before. We can repeat the process to get an unending sequence of numbers
a^2 = 2b^2, b^2 = 2c^2, c^2 = 2d^2, d^2 = 2e^2, .... ,
where a^2 > b^2 > c^2 > d^2 > e^2 ,....
all of these integers being positive. But any decreasing sequence of positive integers must come to an end, contradicting the fact that this sequence is unending. This also provides us with the contradiction to what have been supposed, namely that the square of a rational number is 2. Hence we must conclude that square root 2 is an irrational number.

Daylight saving is the method of adjusting clock so that less work day hours are available in a daytime from sunrise to sunset. Wake up from the bed and see the kitchen clock is set at 8. o’clock am suddenly even when you swear your alarm was set at 7 am. Now you are convinced that you are late at work. But you are not

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