"Science is the source of all other knowledge"
"Mathematics is about numbers and physics is all about mathematical functions"
crazy Quantum mechanics + beautiful relativity = Universe (including us?)
"All that we know from common sense are necessary mistaken or erroneous.. "
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
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 piece 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 the nothing but cosmic music played on the same cosmic strings or membranes.
+ = Everything
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 member of itself. So in either way we come to a contradiction!
This was Russell famous paradox. All mathematics are basically logical deductions. You can follow my website for more logical analysis of mathematics.
The main purpose of this website is to give some basic ideas of classical physics, quantum mechanics, relativity and quantum field theory for those who wants to become a science major or philosopher in science. Theories are at the heart of physical sciences and mathematics. Here most of the major and interesting theories of physics have been discussed. Those who want to understand popular science books will get help from this website. Some electrical and electronics engineering concepts have also been given for undergraduates of electrical engineering. This website is not supposed to include everything. Those who are very curious to understand scientific theories and spacetime physics in somewhat non technical way, will find this website very usefull. So far a little subject matters have been included. In the future more and more topics will be added. Many topics of Mathematics and its applications are also demonstrated in a simple manner. I have already said, all physics is interpreted by mathematical functions. The principle aim will be to justify this claim.
Theories are basically mathematical descriptions of natural phenomena. A theory is like a boat. How well the boat floats gives the soundness of the theory. The better the boat floats, the better the theory is. As long as a theory is able to describe particular phenomena , it is accepted as valid. But it can never be taken as irrefutable and absolute. For a better theory can replace the old one. Such is the case with Newton's universal law of gravitation
There is good news for those who are electrical engineers. The good news is that you guys can be physicists easily than others. If you read this website attentively you will be able to find deep connection between electrical engineering and theoretical physics . Study this website and follow its pages whenever you can.
Quantum Field TheoryFeynman's sum over histories | S-matrix
Theory of relativityRelativity made simple | Special theory of relativity | General theory of relativity | Tensor calculus | Hamiltonian mechanics | Field equation
If you give , you only get in returns...
Quantum mechanicsSchrodinger equation | Matrix mechanics | Dirac equation
In quantum mechanics operators are replacements for classical quantities like position and momentum. They are some kind of functions too.
all the physics in one equation (excluding string theory)
This is the equation which is derived in analogy with Feynman's path integral equation.
There are seven ideas that shook the universe. These are
1. Newtonian mechanics
2. Energy and entropy
3. quantum theory
4. conservation principles
5. copernican astronomy
7. Theory of everything
"we are all here to do what we are all here to do"
Newtonian mechanics made it possible to develop the first mathematical model of the universe. It was a great revolution in science. Everything in science is founded upon principles of newtonian mechanics. It is hard to imagine anything which does not correspond to Newton's law. Quantum mechanics and relativity are developed to account for the phenomena which classical newtonian mechanics fails to explain. But in the limit both theories reduce to newtonian framework. Application of Newton's law is almost everywhere from rocket science to automobile engineering. We have developed an innate sense of the principles of newtonian mechanics in us. We move and throw objects as if we know the position in advance.
On the other hand we can not predict the outcome of quantum mechanics experiment precisely. Principles of quantum mechanics violate our common sense. It is developed using logic and mathematics which defy our intuition. Suppose you throw an electron in the wall and on the wall there are several objects placed. Then there is always some chance that the electron will hit those objects at the same time. This is the kind of weirdness that our commonsense can not grasp. There are other bizzare consequences of quantum mechanics like entanglement and quantum tunnelling. According to Quantum mechanics spooky action at a distance is possible. This faster than light communication had disturbed Einstein too much. According to theory of relativity nothing can travel faster than light and causality is not violated. Quantum mechanics have certainly made many phenomena non-causal. The state of affairs in quantum world can be described with the following example:
You are now seeing an advertisement below served by google. See how it is displayed on your device's screen. Electrons manoeuvre the liquid crystals to block or pass the light from small light bulbs. The light that comes from the display are making this ad look so beautiful. Each pixel on the ad is excited electronically. What you are seeing is only light. Now when you will touch the ad , a set of instructions encoded inside the memory system of your device will be triggered. How will this happen? Well computer has all the informations that are displayed on your device's screen. When you touch the ad , the display will help develop a resistance or capacitance on the specific part of the screen you touch. This will tell the system to take necessary action. Believe it? Now touch the advertisement to see .. if we adopt the theory of radiation, the atom seems to live a "self contained life" - " the world forgotten by the world forgot". But the atom sometimes involves in parcel-post activity. That is to say, it can give a parcel of energy to a postman or it can receive one from him. The postman(who is not a teetotaller) who receives the parcel sways from side to side. The bigger the parcel is, the faster he sways. He runs at the same speed every time whether the parcel is big or small. The postman is the only link between the world of atom and the outside world.
Giving to others is a great virtue...."I know everything is in the mind but mind is a very powerful thing"
Another matter that is of great debate is the problem of free will. According to principles of classical mechanics everything is predictable. Our actions are not governed by us. Every will and volition have previous causes and that cause in turns has another cause. Thus we certainly do not have any free will. But in quantum mechanical world nothing is predictable with absolute certainly and we might possess free will after all. We have freedom to make our own decision. According to classical physics we have only one of two alternatives (yes or no) in our mind. Our choice is predetermined with yes or no option. But according to quantum principles we have both yes and no options, which exist simultaneously in our mind. Thus Quantum mechanics has rescued free will from drowning.
The phenomena of having both yes and no states in our mind is called the principle of quantum superposition. Scientists have been able to find the exact location of the brain where this quantum superposition seems to take place. This part of the brain is called microtubules. There is a close connection between our consciousness and quantum mechanics. Both are mysterious and thus both may be the same.
Dynamics of a mouse
The computer display or screen is a two dimensional greed. Every pixel or point on it is designated a pair of values like coordinate system of theory of relativity. When you move your computer mouse or your finger over touch pad the sensors attached to the mouse of touchpad convert mechanical movement into electrical signals conveying numbers. Transducers are basically used in touch pad . Electrical Transducer produces electricity when pressure and force are applied on it. These numbers that electrical signals carry are the coordinates of the computer display, which tell the system where in the screen the mouse pointer will glow. This is how the mouse pointer of computer works. The process is although interlaced with computer programs.
Have you ever seen the movie matrix? if you like science you must watch this. This movie tells us that our world is in a computer simulation or inside a matrix. what if our world is a complex computer program. We are nothing but the strings of binary digits 1 and 0. It is hard to believe but many scientists believe it to be so. According to the simulation theory our even consciousness is simulated by computer program. I do not know whether this theory is right or wrong but such theory has advantages. By the way here is a clip of the second sequel of the movie."Simplicity is beauty.."
Some of the revolutionary equations of physics in one package
"what is mind never matters and what is matter never mind.."
"We are the children of atoms, so understand the atom if you try to understand the universe"
Electrical and electronics EngineeringPower Engineering | Telecommunication | Control System Engineering | Electronics | Fundamentals of EEE | Digital signal processing | Digital Filters | Maxwell equations | Differential equation and calculus
"God is a great mathematician"
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"
Without speaking about numbers the importance of mathematics can not be described. To describe numbers or define numbers we need to know how to count. To count , in turn, will need three steps : 1) the idea of many-ness 2) one-to-one mapping and 3) sequentially speak or say the number words
First we need to have the idea of many-ness. We need to answer the question of how many? How many objects do we a collection have? Then we map each objects to other objects with one-to-one relation.
Mapping: mapping is assigning each element of a set to exactly one element of another set.
When we assign an element , say our finger to each element of a collection , we speak one for one finger , two for two fingers and so on to last one which will be the cardinal number of our collection or set. Then we give a symbol for the number if we like. This symbol will represent that particular number.
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.
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. 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 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.
So what is the most complicated topic in mathematics? Is it topology, geometry, 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. 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 gives 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.
"When you got nothing , you got nothing to loose.."
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."Things you own end up owning you"
Fractals are geometric objects. Fractal is self-similar object which pattern is never ending - a appearing again and again. For example a snowflake is a
fractal like object which contains similar parts within it. If one looks closer and closer at this kind of fractal like object he will see the same structure
over and over again. Fractals are a great subject matter for mathematicians. We can see fractals almost everywhere. These are in the leaves of trees, in the painting
of an artist and in many other things. These kinds of structure certainly can reveal many secret about our universe. Fractal can be defined using scales.
Fractals are scale invariant. That is these are the same whether you look into smaller parts or bigger parts.
Mandelbrot set is an example of fractal like structure. The equation of Mandelbrot set is the iteration map :
"Mathematics may not teach us how to add love or subtract hate but it ensures every problem has a solution..."
One electron universe
One electron universe is a concept or idea which tells that there are only one electron in the universe. What we observe to be different particles is really one electron going back and forth through time. The world lines traced by a single electron make a twisted structure like a knot in spacetime. Any slice through time represent the single electron at different locations. Positron can be viewed as electron going from future to past in space.
Richard Feynman qouted something which represents this fact of electron going back and forth in time :
" I know why all the electrons have identical mass and charge because they are all the same electron. "
Electron is a point mass which have negative charge. Electrons are the singularities in electromagnetic field. That is to say, they have infinite divergence. What it means need a careful study of Maxwell's equations.
"The universe is the ultimate free launch.."
If you can't beat me then join me..."If you want to reveal the secret of the universe , think in terms of energy, frequency and vibration.."
Duffing equation for harmonic oscillatorsThe equation below contains some interesting terms. First one is the inertia which is the force of Newtonian mechanics. The seconds term is the damping force which is proportional to the velocity. The third term is spring force which is proportional to position of spring . The forth is the non-linear term in x^3 . The right hand side contains a variable force.
What does an equation mean ? Any equation is plausibility of any natural phenomena. According to Ramanujan " An equation has no meaning unless it expresses the mind of God"
The space-time diagram of black hole describes the event horizon and singularity. At the event horizon the light-cones tip over and falls into singularity. The diagram actually describes the sapcetime collapsing of a star when it forms a black hole. When the star shrinks the matter inside the star shrinks too and the density starts to increase.
At some point the density becomes very extreme and event horizon forms. Light wave can no longer escape the horizon. This phenomena is depicted in the diagram above. Our earth can be turned into a black hole of it can be squeezed into a sufficient small size object. Time comes to a halt at the event horizon of the black hole. To an outside observer this time dilation fact will be apparent. Space time diagram above also suggests space and time come to a star.
Miscellaneous pages Algebra | coordinate geometry | Topology for dummies | Symmetry | Bertrand Russell philosophy | Sir issac newton | Topology for dummies | Albert Einstein | Artificial intelligence | Leonhard Euler | Carl Friedrich Gauss | Number theory | what is science
"The best thing about science is that you can apply it to your life"
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"whatever happened, happened for a reason and could not have happened in any other way"
Sigmund Freud was perhaps one of the best experimentalists or clinical psychologists. He used to study and experiment with his patients. He coined the term psychoanalysis. Psycho analysis is the study of mind or it is the theory of how activities and interaction of our mental life is described . Phyche means mind. Psychoanalysis , in general, is concerned with how human behaves. Psycho analysis is based on three apparatus :one is ego, the second is Id, and the third is superego. Id is the uncoordinated instinctual trends, superhero is the critical and moralistic act. Ego plays the role of mediating between id and superego. Freud based the model of psyche with these three apparatus. In terms of these concepts he interpreted the process of dreaming. He also claimed that all sexuality is psycho-sexuality. His theories have impacted modern psychology to a great degree. He concluded some strange consequences from his theories. For example we certainly enjoy pleasure of anal sex from defecation. A child develops sexuality when it sucks its mother's nipple and so on. Full explanation of psychoanalysis needs much space but the basic ideas are those explained above.
Physics of the impossible
Michio kaku have tried to explain all the paranormal activities in the context of science. These are telekinesis , telepathy, mind reading , psychokinesis and many others. We see all these in sci-fi Hollywood movies every now and then. But are these phenomena real or scientific? There is no direct relationship between these and science. Science like to explain stuff in terms of causal relationships. As far as causality is concerned there are no hope for these kinds of phenomena to be true. In science we have explanation and evidences. Paranormal activities have no explanation otherwise these were not called paranormal in the first place.
That does not mean we give up. May be, some future science will explain or make these phenomena possible.
"Without electricity we would not think, your heart would not beat..."