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"Science is the source of all other knowledge"
"Mathematics is about numbers and physics is all about mathematical functions"

theoretical physics equations

Quantum Field Theory

Feynman's sum over histories   |   S-matrix

Theory of relativity

Relativity made simple   |   Special theory of relativity   |   General theory of relativity   |   Tensor calculus  |   Hamiltonian mechanics   |   Field equation

Quantum mechanics

Schrodinger equation   |   Matrix mechanics   |  Dirac equation

Electrical and electronics Engineering

Power Engineering   |   Telecommunication   |   Control System Engineering   |   Electronics  |   Fundamentals of EEE  |   Digital signal processing  |   Digital Filters |   Maxwell equations  |   Differential equation and calculus

String Theory

What is string? string theory explains everything in terms of tiny filament of vibrating energy. This filaments of energy are called strings. String theory is a unified theory or grand unified theory. Sixty years of research has not been experimentally verified yet. Dynamics of strings can be described by two dimensional Lagrangian and a world sheet containing strings :


string theor theory

It is not meant to be easy. But if we follow deductive approach , we can say everything in physics should at least be theoretically related to string theory. How can this be ? All the theories are linked up with string theory just like general thoery of relativity relates almost every aspects of classical to Newtonian physics.
In string theory main ingredient is "Lagrangian" which is a function. It is two dimensional rather than one dimensional lagrangian used in quantum mechanics and general relativity.

"Most incomprehensible matter about the universe is that it is comprehensible"
"equations bring equations"

Understanding Physical science

what is science? Science always begins with myth. First some hypotheses are taken to be a starting point. Hypotheses are all guessworks. Scientists first guess a physical description of a phenomena. Then consequences are drawn from those hypotheses or guessworks. If the experimental results agree with the results it is accepted as a true theory. This is similar to mathematical reasoning but the main difference is that science has to correspond to real world phenomena while mathematics necessarily , does not have to.
Science of electrical engineering focuses on various aspects of one fundamental force of nature , which is electromagnetic force. Electrical and electronics engineering has greatly influenced the development of modern technology. Technological achievements have stunned us. But it is not stopped. Electrical engineering is one of those disciplines that human knowledge has shaped over several hundred years. Nineteenth century was a high time for the physicists and mathematicians, when all the major developments of science took place. Quantum mechanics and photoelectric effect were discovered. Although major inventions were established in nineteenth century , electrical phenomena was known long before that. Theory of Magnetism and electromagnetism were also developed. Electricity and magnetism , together, form theoretical foundation of electrical engineering. Charles-Augustin de Coulomb first formulated the equation of force between two electrically charged particles, which is known as Coulomb's law. Every student of school or college more or less knows about the law. Coulomb's law is very similar to Newton's law of gravity which is still valid for any system of large or massive objects like sun, earth or moon. A simple illustration of Newton's Law of Gravity can be given :

Universal  gravitation
Newton's law of gravity states that force between two object is proportional to their mass and inversely proportional to the distance r between them. Every student of science need to know about Newton's law. It is still used in astronomy and cosmological model. Newton's law can be used to derive Kepler's law of planetary motion. The orbit of the planets are not perfectly circular but rather elliptical. The point closest to the sun is called perihellion and point furthest to the sun is called aphelion. Newtonian physics predicted that perihelion of each planet had to remain fixed. Later we will find that perihelion of each planet does not remain fixed at a point but it shifts by an angle each century.
Universal  gravitation
This is called precession of perihelion of mercury. This is analogous to electrons in eccentric orbit. Electrons in eccentric orbit have to go more than once when they are closest to neucleus while orbiting it. This effect is special relativistic while the former is related to spacetime geometry of general relativity. Coulomb's and Newton's law can both be classified as inverse square law. The inverse square law implies that the force reduces in quadratic manner as the distance increases. For example if the distance is doubled the force will be one-fourth, if tripled , one-nin th and so on. Gauss described the relationship between the charge density and electric field , which tacitly assumes the properties of inverse square law if the surface enclosing a charge distribution is spherical. Even Coulomb's law can be derived using Gauss flux theorem. And back in 16 century an inverse square law was proposed for the light source and its intensity. It seems nature is kind of predisposed to inverse square law and all these laws say something about spherical symmetry. Symmetry results from in-variance when we change our perspective. It can be applied to any shape, objects or many other abstract things. Nature is full of symmetrical shapes. A snowflake shows certain symmetry when we rotate them. Our body has certain symmetry and so have all spherical objects. Symmetry is simplification, which plays a vital roles in physics and mathematics. Whenever possible another chapter will be discussed on symmetry.
I have decided to include Fynman's lectures on gravitation on the field equation page. Here you can get a complete mathematical description of Einstein's general theory of relativity.

conjecture and refutation

This idea was probably coined by Philoshoper Karl Proper. Science progresses through conjecture and refutation. Conjecturing a new phenomena is necessary to develop a theory. Scientists have to observe a lot of phenomena before give a hypothesis for a new theory. This provess is called generalization. Inductive reasoning plays a similar role like this. Scietists gather a vast number of data to find a new theory. Thus the conjecturing process is done. Science depends on conjecturing new hypotheses which serve as the model for a new theory. That does not mean deductive reasoning does not apply to developin a new thoery. Einstein showed how deductive approach can be too powerful tool. General theory of relativity is a patchwork of logic and experimental results. When a theory has been developed scientists must test it before regarding as valid.
In Feynman words : " No matter who you are, how smart you are if the experimental test does not confirm your hypothesis , your theory is wrong"
The theory or physical law is universal. It should apply to everywhere. But it does not mean it will be valid always. We can only get closer and closer to truth. The validity of physical laws depend on the experimental results or empirical evidences. If one instance of those is found, that contradicts the fact that the law demands, then the theory must be refuted. A new theory must be proposed to replace the old one. Such was the case with Newton's Law of universal gravitation. Einstein's theory of general relativity is more elegant and precise than Newton's law. Usually new relations enter into the mathematical formula and a refinement of the law is processed.

"If you want to be a physicist , you have to do two things, first learn mathematics, secondly do the same"

"abstraction is the real source of power"

Applied vs pure mathematics

Computer Science is another revolutionary branch of engineering science , which is directly related to applied mathematics. Semiconductor devices led the rapid development of computer technology. Computer science and electrical engineering are contingent and complementary. Practical computer would not be feasible without applications of electrical and electronics engineering . Any electrical or computer engineer would not hesitate to admit the fact. But at present there is a race between the two and in many cases computer science has outpaced electronics and electrical engineering. Information technology is now a dominating field. Nothing has connected the world more than the internet has in recent years. The information age actually began when Claude Shannon formulated his famous equation relating channel capacity to channel bandwidth and signal-to-noise ratio. The formula is simple enough to be specified :
information theory
Where B is the specific bandwidth of the channel. But the equation describes a communication related phenomena, which is thus a topic that falls under the scope of electrical and electronics engineering. Claude Shannon was a mathematician too. Mathematics is at the heart of telecommunication like all other fields of science. Without mathematical equations no laws of physics or phenomena can be explained quantitatively. So formulation with mathematical equations is necessary. There are seventeen equations that have changed the world.
17 equations
One among these has created a revolution in atomic physics. It is Schrodinger equation. Schrodinger equation finds the wave function ψ(x,t) of a particle like electron. Atomic orbitals of Hydrogen have been perfectly described wave function ψ(x,t). The importance of Schrodinger equation is seen in its extensive use in Quantum electrodynamics and quantum field theory.
The second most revolutionary equation is the above list is the wave equation. It is a second order differential equation which describes how waves propagate in space and time. Do not confuse it with spacetime of theory of relativity.
Third one is the energy mass equivalence relation. It can be derived using formula of special relativity. The most devastating application of this formula was the development of atomic weapons.
The next one is the entropy formula. Entropy is energy divided temperature. That is how much energy is being lost as unavailable energy is entropy in simple sense.
Maxwell's equations are equations that have unified electricity with magnetism. It is perhaps one of the most elegant mathematical formalisms in theoretical physics. The development of advanced theoretical physics had been accelerated by Maxwell's equations. The next important and innovative equation in the list is the equation describing chaos theory. It is a difference equation. There is difference between difference equation and differential equation. Difference equations involve the value of function to its earlier value distinguished by two consecutive integers.
Joseph Fourier developed his theory on function that any periodic function can be represented as a sum of infinite number of sines and cosines. And in addition a function can be transformed into another domain by particular integral formula.

Mathematics provides a model for the real world phenomena and simplify many problems related to them. Electrical engineering is dependable on applied mathematics rather than higher level abstraction used in pure physics or mathematics. So mathematics is concerned with problem solving in engineering practices and its applications. In order to solve the problems , we use the mathematical formula that best explains the phenomena. Various mathematical formulas of algebra, calculus, geometry and trigonometry are applied in solving the problems. The variables and parameters studied must correspond to real world application. In particular these must quantify the properties of the engineering system. So applied mathematics is always concerned with system modeling, problem and puzzle solving related to real world applications. The variables and parameters used in the electrical circuit analysis can be given definite value by applying measurement.


it is impossible for us to think of any thing, which we have not antecedently felt, either by our external or internal senses


On the other hand pure mathematics is the mathematics for its own sake. Pure mathematician develops theorems and focus on proving it. It is totally abstract reasoning. Pure mathematics deals with patterns , logic , propositions, theorems and argument. In pure mathematics there are statements or propositions which may contain variables. When these variables are given specific value the statements becomes specific and part of applied mathematics. Let us think about a statement of pure mathematics : for any a, b and c , if they are the base , vertical and hypotenuse of a right triangle respectively then a(squared) + b(squared) = c(squared). a, b and c can be anything and this statement is always true as long as they are the sides of any right triangle. But when we substitute a for real power and b for imaginary power and c for apparent power we get a specific right angle triangle whose base , hypotenuse and vertical sides real, apparent and imaginary power. We thus step into the realm of applied mathematics. All that we need are the numbers to be used as length of the sides of that triangle. If we know real and imaginary power we can find the apparent power and vice versa. The stated proposition containing " if (proposition) then (proposition)" is an example of material implication. Material implication ( if-then) is a relation between statements which hold true when one statement logically follows from one or more statements. "P implies Q materially" means that either P is false or Q is true. In this regard false proposition implies every propositions and a true proposition is implied by every proposition. On the other hand formal implication is more general. Formal implication is the relation of inclusion to obtain from the logic of predicates of the form (x) { if 'x is F ' then 'x is G'} . For example, if the two predicate functions stated are ' x is a man' and 'x is mortal' , we can infer that all men are mortal. We now have a new relation of inclusion. Thus formal implication is a single proposition. It is not a relation of two propositions. Material implication is a limiting case of formal implication when the variables in the formal statements are replaced with specific value. It is concerned with the existing state of affairs. These two kinds of implication namely material and formal implication, are vital to mathematics for all kinds of deduction.
Pure Mathematics is the class of all propositions of the form “p implies q”, where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.
Propositional functions are also statements which does not have the truth value of true or false except when the variables in it are given specific value. Premise is a proposition which can be either true or false as opposed to axioms which is assumed to be true. In pure mathematics , premise is used to draw a conclusion which dictates the premise has to be true. There is other rules of logic like conjunction, dis-junction, negation, contra -positive, principle of importation and few others. The implication ( material and formal) is what describes the essence of mathematics. Mathematics, in this way, can be interpreted as "if such and such thing is true of something then such and such other things is also true of that thing". It is irrelevant to mathematics what such thing really means or what such things are.
Pure mathematics has logical constant which is not the same as usual constant in mathematics. All logical constants can be defined in terms of implication or relations. Logical constants are the forms that does not change when we vary other terms in the proposition. Term can be anything which occur in mathematical statements. So terms and relations constitute the general form of propositions. We can get different instance of the same proposition by varying the terms in that proposition. The relation remains the same , which can be termed logical constant. All the theorems and truths of mathematics can easily be established by using few premises and some rules of deductions. This is how logic has imparted(given) a solid foundation to mathematics although the conversion can be difficult and too abstract. But reduction of mathematics to logic is one of the greatest achievements human mind have ever made. Bertrand Russel and Alfred North Whitehead had established a rigorous foundation of mathematics using logic in their famous Principia Mathematica. They came up with a definition of cardinal number with pure logic :
"Cardinal number of a class is the class of all classes that are similar to it."


Cardinal number is the size of some class. By size it is to mean how many elements the class contains. This word "cardinal" is probably due to Cantor. On the other hand ordinal number is the number which identifies series. More generally it is class of generating relations of series. One of the familiar ordinals is ω. It is the class of all infinite series. Since the definition do not presuppose number, it is better to give this type of series a name. We call it class of progressions. The definition in words would be :
The ordinal ω is the class of all classes u such that there is a one-to-one relation R such that u is contained in the domain of R, the class of terms to which different u has the relation R is contained in u , without being identitcal with u and which, if there is a class s to which belongs at least one of the terms of u , to which any u does not have the relation R and to which belongs all the terms of u , to which a term of the common portion of s and u has the relation R , then u is contained in s.
We can define progression similarly using the definition "relation" : A progression is a one-to-one relation such that there is just one term belonging to the domain and not to the converse domain and the domain is identical with the posterity of this one term.

Ordinal arithmetic

Ordinal number as defined is a very interesting concept. It has lot of amazing property. Unlike cardinal numbers it follows different rules of addition and multiplication. Various kind of ordinal numbers can be represented using a chart.
ordinal numbers
The first ordinal number that comes after all the natural numbers 1,2,3 , ... is the first transfinite ordinal ω. It is the class of all infinite series N. The next ordinal after ω is ω + 1. It is not the same as 1 + ω. It is different . The first ordinal number that comes after all the natural numbers 1,2,3 , ... is the first transfinite ordinal ω. It is the class of all infinite series N. The next ordinal after ω is ω + 1. It is not the same as 1 + ω. It is different. The latter is the same as ω. If you add 1 before the series of ω you still get the same series. But if you add 1 to the end of series defined by ω you get a different series. That is the reason for the difference.
If you continue to add numbers to ω like 1, 2, 3,.. you end up with ω + ω = ω.2 . Then you can add numbers to ω.2 and end up with ω.3. After continuing this process with ω you finally get ω^2 (squared). If you do this repeated with various powers of ω you will end up with another transfinite number ω to the power ω or ω^ω. This is quite extraodinary and Cantor had revolutionalized mathematics of transfinite numbers in a way that can never be superseded.

Number is a concept which have surprised many philoshopers and mathematician for centuries. Numbers are of many kinds. They can be rational numbers, irrational numbers, prime numbers, transcendental numbers and many others. Prime numbers are still a mystery. Euclid first proved that there are infinite number of prime numbers. He proved it by using proof by contradiction.


Suppose there is a list of primes denoted by p(1), p(2), p(3),...p(n) where n is finite integer. We now form a new number q = p(1)p(2)p(3)..p(n) + 1; This number must be divided by a prime. Let it be m. If m is one of the prime listed above then q- p(1)p(2)p(3)..p(n) must be some number other than 1. Two This is a contradiction. So we must conclude that the list is incomplete. No matter how long the list is there is another prime not in the list. Let 1, 2, 3 and 5 be our list. Then q = 30+1 =31. This number 31 is a prime itself , which was not in our original list. This proves our hypothesis.
In the above proof we used fundamental theorem of arithmetic , which states that every composite number is a product of other prime numbers. 12 is a composite number which is multiplication of 3 and 4 , which are prime numbers.
One of the most historical numbers is irrational number. Irrational numbers are the number which can not be expressed as a ratio of two intergers. The theory of the irrational is a lengthy subject. The history began with the incommensurable quantities. Incommensubale quantites can not be simply multiples of each other. They do not share any common measure of length. The diagnal of a square which side is one is root(2) which Greek found to be not integer multiple of some other length (unit in this case). This kind of incommensurable quatities are called irrational numbers. So irrational number was found and needed to be defined. Dedekind first defined irrational numbers in a very intuitive way. He defined the irrational using a division of real numbers known as Dedekind cut. There are three properties which must be satisfied to make his theory useful.
1. If a >b and b>c then a>c . Between any two numbers a and c there is another number b. Suppose there are two numbers 2 and 5 in the real number line. Then we can find another number by taking the average of these two ([2+5]/2 = 3.5 ). Using geometric notion we can say that b lies between a and c.
2. there are an infinite number of numbers between any two real numbers.
3. we can always partition the real number line in two sets(A, A`) such that all numbers of A is less than all numbers of set A` and the first set A has a greatest number while the second set contains no least element.
In such a case the partition (A, A`) will define a cut known as Dedekind cut. This cut will represent an rational number .
Now suppose the first set A contains no greatest element while the second A` contains no least element. In such a case the partition (A, A`) will define a cut known as irrational Dedekind cut. Suppose there is a number D such that x(square) < D < (x+1)(square) where x is any other number whose square is less than D and the square of x added to one is greater than D. So root(5) = 2.2360.. is an irrational number according to our definition. We can always find a x such that the conditon is fulfilled. For example x=2.2 if squared gives 4.92 and 3.22 (square) = 10.32 is greater than 5. The square root of 5 is between two whole numbers 2 and 3. We can increase the value of x without any bound like 2.23, 2.234, 2.235, 2.2355, and so on as long as its square its square is less than 5. The other set contain no least element as we can choose x to be 2.60001, 2.236001, 2.23601, 2.23602, 2.236022, 2.23612 all of which are greater than 2.2360.. by a small amount. Thus our condition is fulfilled and we get a cut by the number root(5) which is according to our definition is irrational cut. Dedekind thus defined irrational numbers using only rational numbers. But it is mistaken belief that irrational number must always be the limit of rational numbers.
We all know about Pi (π). It is an irrational number. It is the ratio of a circle's circumference to its radius. It is such a number which have got attention of many mathematicians and philosophers. It is approximately equal to 22/7. The decimal portion has no ending. Neither it has any pattern that repeats itself. It fascinates many with its illusive characterstics. Powerful computers are now used to calculate the vast number of digits that it has as an unending series.
is value of π always constant? It is constant when we consider all the circles in plane geometry. The ratio of their circumference to their radi will always turn out to be 3.141.. But when we have a curved space this is not the case. This ratio will not be constant. Here is more details on field equation
Transcendental numbers are those numbers which are not solutions of algebraic equations. Algebraic equations are all the equations of the form P=Q where P and Q are polynomials. The solutions of algebraic equations are called algebraic numbers. These algebraic numbers include rational numbers , irrational numbers , e , π and many others.
Measurement can be done when superposition of two magnitudes is possible. By "superposition" we simply mean the coincidence of two things. But it has other definition in mathematics and physics. Measurement is an integral part of modern physics and engineering. By performing measurement we attribute (assign) units to certain physical quantities like scalars and vectors. In engineering there are standard units for every quantities measured. Measurement and standard of units are all maintained by convention. Conventional system is like keeping a name of a child when it is first born. The name serves as a standard by which we all know the child. There is no reason to suppose that all the systems of measurement and units are absolute or one measurement is more valid than the other. Measurement follows magnitude. Magnitude can be defined when we have a general conception which is capable of various determinations. Any set of such different determinations is called a manifold. Manifold can be continuous or discrete based on whether the passage from one determination to another is continuous or discrete. Magnitude is the relative size of things by which we can compare those things. A scalar quantity has only magnitude whereas vector quantities have both magnitude and direction. These are very useful entities in the field of both engineering and science.

Philosophy

I like to share some of own thoughts. This might be regarded as General philosophy. A great amount of the philosophical ideas has been written by Bertrand Russell. He was a great philosopher , whom we all know very well. Philosophy has the freedom of speculation and free thinking. I will now enjoy such freedom :
My first issue is with the mind -body dualism which is also known as monadism. According to such concept there are two distinct entities in this world. One is mind and another is body. These are also equivalent to mind and matter. Now if we take our mind as the world of abstract thoughts, then we should not claim mind-body dualism to be correct. If we move from one place to another place our mind also moves. That means our mind or abstract thoughts are also matter which can only move through space and time.
A great deal of Greek philosophy can be stated. Aristotle probably started the study of matter and motion. He thus first informally developed the field of physics. Although there is controversy about who first invented physics. Physical universe is the material universe that we inhabit. Physics is the study of this material universe and its laws.
Lots of philosophical ideas has provided a firm ground for the modern theory of the universe. One the Greek thought that earth was the centre of the universe and the solar system was all that was inside the universe. This was so called geo-centric universe. Then helio-centric model of the universe was developed. This was the idea that sun was at the center of the universe. Later kepler and Newton gave mathematical description of the planetary motion. This is how scientifc model of the solar system was established. Newton's universal gravitation seems to be found to apply everywhere is the universe and scientists were able to bring the whole universe in the scope of physics and mathematics. Newton developed his first masterpiece "principia mathematica" describing many other physical phenomena.
Galileo Galili was another great scientist whom we all know very well. He studied speed, motion, projectiles and many more. His experiment in the inclined tower of pisa is well known to us. He first postulated the principle of relativity. This principle states that laws of physics are unchanged by the uniform motion in a straight line.

Penrose's lectures

The spacetime of Einstein’s general relativity Finally, we come to the Einsteinian spacetime E of general relativity. Basically, we apply the same generalization to Minkowski’s M, as we previously did to Galileo’s G, when we obtained the Newton(–Cartan) spacetimeN. Rather than having the uniform arrangement of null cones depicted in Figure below, we now have a more irregular-looking arrangement like that of Fig. 17.17. Again, we have a Lorentzian (+ ---) metric g whose physical interpretation is to define the time measured by an ideal clock, according to precisely the same formula as for M, although now g is a more general metric without the unifomity that is the characteristic of the metric of M. The null-cone structure deWned by this g specifies E's causality structure, just as was the case for Minkowski space M. Locally, the differences are slight, but things can get decidedly more elaborate when we examine the global causality structure of a complicated Einsteinian spacetime E.


spacetime

spacetime

An extreme situation arises when we have what is referred to as causality violation in which 'closed timelike curves' can occur, and it becomes possible for a signal to be sent from some event into the past of that same event! See Figure. Such situations are normally ruled out as 'unphysical', and my own position would certainly be to rule them out, for a classically acceptable spacetime. Yet some physicists take a considerably more relaxed view of the matter being prepared to admit the possibility of the time travel that such closed timelike curves would allow. On the other hand, less extreme though certainly somewhat exotic—causality structures can arise in some interesting spacetimes of great relevance to modern astrophysics, namely those which represent black holes.
we encountered the fact that a (pseudo) metric g determines a unique torsion-free connection = for which ∇g = 0, so this will apply here. This is a remarkable fact. It tells us that Einstein’s concept of inertial motion is completely determined by the spacetime metric. This is quite different from the situation with Cartan’s Newtonian spacetime, where the '=' had to be specified in addition to the metric notions. The advantage here is that the metric g is now non-degenerate, so that = is completely determined by it. In fact, the timelike geodesics of = (inertial motions) are Wxed by the property that they are (locally) the curves that maximize what is called the proper time. This proper time is simply the length, as measured along the world line, and it is what is measured by an ideal clock having that world line. (This is a curious 'opposite' to the 'stretched-string' notion of a geodesic on an ordinary Riemannian surface with a positive-deWnite metric;. We shall see,, that this maximization of proper time for the unaccelerated world line is basically an expression of the 'clock paradox' of relativity theory.)
The connection ∇ has a curvature tensor R, whose physical interpretation is basically just the same as has been given above in the case of N. The causality structure of E is determined by g (as with M, see Figure ), so extreme unphysical situations with 'closed timelike curves' might hypothetically arise, allowing future-directed signals to return from the past. What locally distinguishes Minkowski's M, of special relativity, from Einstein's E of general relativity is that R = 0 for M. In the next chapter we shall explore this Lorentzian geometry more fully and, in the following one, see how Einstein's field equations are the natural encoding, into E’s structure, of the 'volume-reducing rate' 4πGM . We shall also begin to witness the extraordinary power, beauty, and accuracy of Einstein's revolutionary theory

Density of states

In solid state physics and condensed matter physics density of states is the number of available states that can be occupied by a system at each energy level. It is usually described by probability density function and it generally an average over all space and time domains of the various states occupied by the system. In general the density of states, related to volume V and N countable energy levels, is defined by:

density of states
Where k is some parameter related to the wavelength by the relation k = 2π/&lamda;
density of states

Negative frequency

The concept of negative frequency comes from the Fourier decomposition of any periodic function. It is also very closely related to Laurent series description of any function. It is the frequency of negative power terms in the Laurent series. In Riemann sphere this splitting of frequency is possible.
The coordinates z and w (= 1/z) give us two patches covering the Riemann sphere. The unit circle becomes the equator of the sphere and the annulus is now just a 'collar' of the equator. We think of our splitting of F(z) as expressing it as a sum of two parts, one of which extends holomorphically into the southern hemisphere—called the positive-frequency part of F(z)—as deWned by F+(z), together with whatever portion of the constant term we choose to include, and the other, extending holomorphically into the northern hemisphere—called the negative-frequency part of F(z)—as defined by F+(z) and the remaining portion of the constant term. If we ignore the constant term, this splitting is uniquely determined by this holomorphicity requirement for the extension into one or other of the two hemispheres. It will be handy, from time to time, to refer to the 'inside' and the 'outside' of a circle (or other closed loop) drawn on the Riemann sphere by appealing to an orientation that is to be assigned to the circle. The standard orientation of the unit circle in the z-plane is given in terms of the direction of increase of the standard y-coordinate, i.e. anticlockwise. If we reverse this orientation (e.g. replacing y by -θ), then we interchange positive with negative frequency. Our convention for a general closed loop is to be consistent with this. The orientation is anticlockwise if the 'clock face' is on the inside of the loop, so to speak, whereas it would be clockwise if the 'clock face' were to be placed on the outside of the loop. This serves to define the 'inside' and 'outside' of an oriented closed loop. Figure below should clarify the issue
Riemann sphere frequency splitting
This splitting of a function into its positive- and negative-frequency parts is a crucial ingredient of quantum theory, and most particularly of quantum field theory, as we shall be seeing later. The particular formulation that I have given here is not quite the most usual way that this splitting is expressed, but it has some considerable advantages in a number of different contexts (particularly in twistor theory). The usual formulation is not so concerned with holomorphic extensions as with the Fourier expansion directly. The positive- frequency components are those given by multiples of e^(-inx), where n is positive, as opposed to those given by multiples of e^(inw), which are negative frequency components. A positive-frequency function is one composed entirely of positive-frequency components.
^ is here the exponential
However, this description does not reveal the full generality of what is involved in this splitting. There are many holomorphic mappings of the Riemann sphere to itself which send each hemisphere to itself, but which do not preserve the north or south poles (i.e. the points z = 0 or z=1). These preserve the positive/negative-frequency splitting but do not preserve the individual Fourier components e^(-inw) or e^(inw). Thus, the issue of the splitting into positive and negative frequencies (crucial to quantum theory) is a more general notion than the picking out of individual Fourier components.
In normal discussions of quantum mechanics, the positive/negativefrequency splitting refers to functions of time t, and we do not usually think of time as going round in a circle. But we can use a simple transformation to obtain the full range of t, from the 'past limit' t = -&nfin; to the 'future limit' t = ∞, from a w that goes once around the circle—here I take x to range between the limits w = -π and w = π (so z = e^(ix) ranges round the unit circle in the complex plane, in an anticlockwise direction, from the point z = -1 and back to z = -1 again; see Figure below). Such a transformation is given by
Riemann sphere frequency splitting
In quantum mechanics, positive/negative-frequency splitting refers to functions of time t, not assumed periodic. The splitting of Fig. 9.5 can still be applied, for the full range of t (from -∞ to +∞) if we use the transformation of relating t to z(= e^ix), where we go around unit circle, anticlockwise, from z = -1 and back to z = -1 again, so x goes from -π to π
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"The universe is a gigantic machine run by rules"

Bernaulli's principle

Bernoulli’s principle formulated by Daniel Bernoulli states that as the speed of a moving fluid increases (liquid or gas), the pressure within the fluid decreases. Although Bernoulli deduced the law, it was Leonhard Euler who derived Bernoulli’s equation in its conventional form in the year 1752.
Bernoulli’s principle can be derived from the principle of conservation of energy. This states that the total mechanical energy of the moving fluid comprising the gravitational potential energy of elevation, the energy associated with the fluid pressure and the kinetic energy of the fluid motion, remains constant.

The Bernalli's equation relating to fluid volume and pressure is

Bernalli's equations

Euler equation

Once a person asked Euler "can you prove that God exist?" Euler replied that he could. He told the man his famous equation

Euler equation
Hence he claimed that God existed. All the mysterious numbers are in one equation. These are one(1), Pi(π), imaginary(i) and e .

Geometry

Geometry is the highest excercise of human mind. Geometry is the study of shapes and figures in general sense. It deals with space. To define geometry properly we have to speak of Dynamics. Dynamics is the study of matter which is subjected to , and as caused by motion also affected by and exterting force. Thus Dynamics needs to describe the change of position of the matter and its configuration in space. This compels us to use a mathematical framework which can be called geometry. Geometry must exist before Dynamics do. To say that geometry depends on matter is a gross error. But we will see that some matter involves in geometry. To study space properties of matter, is reduced to its bare minimum. This matter is very abstract and and wholly different than matter of Dynamics. In geometry we avoid entirely the category of causality of causation. , so essential to Dynamics and retain nothing but only spatial adjectives. The kind of rigidity affirmed of this abstract matter- a kind which suffices for the theory of our sciences, though not for its application to the objects of regular life- is purely geometrical and asserts no more than this: that since our matter is devoid of causal properties, there remains nothing, in empty space, which is capable of changing its configuration in empty space.
Euclidean geometry was first developed by Euclid and gave us the idea of points, lines , planes and other entities. It was the first geometrical system ever developed. Euclid employed some notions and axioms. Many propostions and corollaries were latter deduced from those. For example Euclid's fourth proposition states : "If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides."
Euclid also stated some notions in addition to the axioms. All the truths and consequences follows from those five axioms and notions stated in his book "elements". Here are the notions used in his geometry
: Things that are equal to the same thing are also equal to one another (the Transitive property of a Euclidean relation).
If equals are added to equals, then the wholes are equal (Addition property of equality).
If equals are subtracted from equals, then the differences are equal (Subtraction property of equality).
Things that coincide with one another are equal to one another (Reflexive Property).
The whole is greater than the part.
Modern scholars agree that Euclid's postulates do not provide the complete logical foundation that Euclid required for his presentation. Modern treatments use more extensive and complete sets of axioms. Modern treatment is based on logic. All of the fundamental mathematics can now be reduced to logic. That all mathematics are deductions from logical principles by logical principles is one of the greatest discoveries in science. Numbers , for instance, are defined by peano in pure logical terms. In defining any mathematical system , we need to have some hypotheses. Using rules of deductions we then draw consequences of the hypotheses. Euclidean propositions follow from Euclidean axioms. That is the logical interpretation of Euclidean geometry if we are unwilling to assume the axio ms to be true without proof. Axioms are what we assume to be true. But is there a logical reason for doing that? What if we deny the validity of the axioms? That is exactly what happened in case of parallel postulate. Denying this axiom we have got non-Euclidean system. Both Euclidean and non-euclidean system are equally true. They both assert implications. For example if P is true of some entities x, y , z then Q is also true of the same entities. We neither assert P nor Q separately. We only assert certain relation between P and Q. This relation is called implication. Thus in mathematics we don't really know what we are talking about neither we know what is true. Statement like " 1 + 1= 2 "can be interpreted as the propositional function " if x is one and y is one and x is not y then x and y are two" . This is a proposition of arithmetic and is always true for any x and y. We do not know what x and y are to be but if they have the property stated in the proposition stated above then the statement is always true. This also gives us logical addition of two classes namely class x and class y. Suppose two classes x and y have one terms each . Then we can form a class by taking those terms of class x and class y. Now if we choose a term from the resultant class it will always belong to either class x or class y. What we get is the logical sum which is itself a class of terms which belong to either class x or class y. Or if a is not b , whatever x may be x is a c is always equaivalent to x is an a or x is a b. This is how all references to numbers are gone.

Lie Derivative

Parallelism in general relativity geometry is path dependent. for a connection ∇, I have been expressing things using the physicist’s index notation. In the mathematician’s notation, the direct analogues of these particular expressions are not so easily written down. Instead, it becomes natural to follow a slightly different route. (It is remarkable how differences in notation can sometimes drive a topic in conceptually different directions!) This route involves another operation of differentiation, known as Lie bracket—which is a more general form of the operatio. This, in turn, is a particular instance of an important concept known as Lie derivative. These notions are actually independent of any particular choice of connection (and therefore apply in a general unstructured smooth manifold), and it will be pertinent to discuss the Lie derivative and Lie bracket generally, before returning to their relevance to curvature and torsion at the end of this sectionFor a Lie derivative to be deWned on a manifold M, however, we do require a vector Weld j to be pre-assigned on M. The Lie derivative, written £(η) , is then an operation which is taken with respect to the vector field η. The deriative £ η Q measures how some quantity Q changes, as compared with what would happen were it simply 'dragged along', by the vector field η. See Figure below. It applies to tensors generally (and even to some entities different from tensors, such as connections). To begin with, we just consider the Lie derivative of a vector field ζ (=Q) with respect to another vector field η. We indeed find that this is the same operation that we referred to as 'Lie bracket' , but in a more general context. We shall see how to generalize this to a tensor field Q afterwards.
lie derivative
Recall that a vector field can itself be interpreted as a differential operator acting on scalar fields F, C, . . . satisfying the three laws (i) ζ(ψ + φ) = ζ(ψ) + ζ(φ), (ii) ζ(ψ φ) = ζ(ψ)φ + ψ ζ( φ) and (iii) ζ(k) = 0 if k is a constant. It is a direct matter to show that the operator v, defined by
lie derivative
satisfies these same three laws, provided that η and both ζ do, so w must also be a vector field. The above commutator of the two operations η and ζ is frequently written in the Lie bracket notation
lie derivative bracket
The geometric meaning of the commutator between two vector fields η and ζ is illustrated in Fig. 14.14. We try to form a quadrilateral of 'arrows' made alternately from η and ζ (each taken to be O(e) ) and find that v measures the 'gap' (at order O(e^2) ). We can verify that commutation satisfies the following relations
lie derivative bracket
just as did the commutator of two infinitesimal elements of a Lie group.