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

Theoretical physics math

# 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 :

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 :

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.

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.

# 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 :

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.

One among these has created 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.
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.

### 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

### 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

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 and what remains are the variables and forms. we can also define logical product of two classes. It is the class of terms belonging to every class in which the two mentioned classes are contained.
Thus for example "Euclidean Geometry as a branch of pure mathematics, consists wholly of propositions having the hypotheses "S is a Euclidean space". If we go on to: The space that exists is Euclidean", this enables us to asserts of the space that exists all the consequents of all the hypotheticals constituting Euclidean geometry. Now the variable S is replaced by the constant "actual space". But by this step we pass from pure to applied mathematics".
Everything in the physical science from law of gravitation to building bridges, from spectroscope to navigation would be profoundly modified by the considerable inaccuracy in the assumption that our space is Euclidean.

### Mathematical Induction

An useful principle of pure mathematics is mathematical induction. It is used to prove a lot of theorems and mathematical statement. If 0 has a property and n+1 also has it, then if n has this property then every natural number has this property. That is, it belongs to all the natural numbers. Whenever we define 0 we can get to any natural number in a finite number of steps by successive addition of 1 to it. This mathematical induction is only valid for finite numbers so we can get to the number by successive addition of one. So natural number is sometimes called inductive number also. It is not applicable to infinite numbers which are similar to proper part of itself. An example of such infinite number is first transfinite cardinal ℵ (0). If we remove one or several terms from it the cardinality still remain the same. It (ℵ0) is the the cardinality of all the natural numbers. Mathematical induction can be properly called a method or law of mathematics. By this principle we go from particular to general cases.

### Electrical machines and equipment

Transformer testing and Electrical machinery are some of things that electrical and electronics engineering is concerned with. Synchronous or asynchronous motors and generators fall in the scope of electrical machines engineering. As the name suggests, the synchronous machines run at a constant speed called synchronous speed. They are useful in many cases when constant speed is expected. They maintain a constant speed irrespective of the mechanical load connected to the rotor. Synchronous motor is not self starting inherently. When it is connected to the supply voltage it does not get necessary torque to rotate. So some kind of starting mechanism is needed to start synchronous motor. On the other hand induction motor does not rotate at synchronous speed. The synchronous speed is the speed of the rotating magnetic field inside motor. Rotating magnetic field is the resultant of all the separate out-of-phase magnetic fields that the stator poles create.

The rotating magnetic field is a vector or phasor whose frequency is proportional to the frequency. There is a precise mathematical relationship between the synchronous speed and line voltage frequency. It is 120f/P where f= frequency, P= number of poles. Induction motor is started and rotated by the principles of induction. Induction motor can not run at synchronous speed. The deviation from the synchronous speed provides necessary torque to rotate the rotor . So there should be a non zero relative speed between the rotor and rotating magnetic field. The induced emf (electro motive force) nin the rotor produces current in the rotor coil. The current in turn produces the necessary torque. If there were no relative speed there would be no emf generated, hence no torque. The rotor always try to catch up with the rotating magnetic field but could not reach synchronous speed for the reason mentioned above. The numerical difference between the synchronous speed and rotor speed is called slip. Speed of induction motor vary according to the load attached to them. The induction motor when seen as an application of electromagnetic induction is nothing but a transformer. We can derive an equivalent circuit for the induction motor too.

Generators are the reversed operation of synchronous motors. In the field winding of generators dc voltage is supplied. Thus a strong magnetic flux is created to excite the rotor winding. A prime mover is used to rotate the rotor at constant speed. The flux created by the rotor field cuts the stator winding and generates desired power. So a kind of energy conversion takes place. Prime mover provides mechanical energy which is converted to electrical energy. Generators are the most important component of a power plant where electricity is generated. There can be many type of power plant like hydro-electric, gas or oil driven plant. In almost every power plant there may be one or several generators operating at the same time. This is to meet the growing demand of electricity.

this website is about theoretical physics and mathematics, theory of relativity general, theories of phyiscs

In heat engine energy conversion takes place between heat and mechanical work. Laws of thermodynamics governs such system. The zeroth law states that if two systems A and B are in equilibrium with a third system C then they are in equilibrium with themselves. First law implies that perpetual machine is not practically implementable. The internal energy of a closed system always changes when some work is done on the system or heat is supplied to it. The mathematical formulation states:

Where W = work done one the system ,
δQ = Total heat
δU = internal energy
So it follows that total internal energy is some kind of mixture of external energy and heat energy. We can not get anything other than this energy from the system. The total energy is always conserved. Perpetual machine is not possible according to this law of thermodinamicaa..
The most important of those laws is the second law which says entropy of closed or isolated system can never decrease over time. The universe is heading toward a state of maximum entropy. Let us try to explain what entropy really is. Heat is the by product of any process no matter how efficiently the process runs. So heat is considered the worst form of energy. it degrades all forms of energy. That's why entropy has been defined in terms of heat energy. It is actually the amount of lost heat that can not be recovered. It is impossible to construct a heat engine whose sole purpose is to convert heat to work, which is an alternative statement of the second law. So we are losing heat energy every moment and it is happening spontaneously. When all the heat energy is lost , the entropy will be maximum. Entropy can be formulated as below :

Small incremental change of heat per unit temperature in irreversible cycle is less than small entropy change S which is zero. The total change of this quantity turns out to be negative. As a result the the entropy of environment increases . On the other hand the change (dQ/T) of reversible cycle is the same as the system entropy ds. If we integrate these two values over a complete thermodynamic cycle , both terms of the equation will be zero. As a result the entropy of system and environment does not change. This is also called Clausius formulation of entropy , which says for a cyclic process the integral of the quantity (dQ/T) always equals zero or less than that.

∮dQ/T <= 0
Greatest scientist Stephen Hawking proved that Black hole has entropy. That means black hole has temperature and it radiates energy. He theorized a mathematical relationship between black hole temperature and its surface area (A).

Radiation means energy and energy means mass which Einstein's showed by his famous equation (E=mc.c). So Every black hole will be extinct with a pop. The time required for such a process is very long though. Entropy can be defined in terms of disorder that always increases over time.

This is an indication that heat always flows from hot body to cold body. If it is to be reversed, work has to be done on the system. Boltzmann formulated the entropy equation based on any system's microscopic configurations. The basic idea is that nature evolve toward the state that is more probable than others. We see a egg falling from a table and getting scattered on the floor. We do not see the opposite : scrambled egg getting back on the table and reassembled into its original form. This is highly unlikely and if we could wait for sufficient amount of time(probably the age of the universe) we were able to see that happen. Universe will ultimately face heat death and no process will continue any more. We do not need to worry at the moment. That will not happen any time soon. Our sun will still be active for the next few billions of years from now. And there are countless number of other stars in the universe to die out. The concepts of entropy and second law are extremely crucial in physics and scientific development. Physicists are aware of the fact that if their scientific law violates the second law then there is no hope for it.

Use of electrical machinery is pervasive all around the world. They are useful because they can do physical work or can convert energy into another form. They save a lot of human work and time. Modern day machines are now more sophisticated than those of early age. This have been possible because of advanced technology. Machines are becoming more automatic day by day. Electrical machines are driven electrically. Most of electrical machines deliver mechanical power. But there is some difference between electrical and mechanical system. This distinction is necessary to understand although there is an analogy between them. We can come up with the same differential equation by replacing the analogous variables for each other. An analogy can be found between mass -spring system and R-L-C circuit. And we have a number of parameters like force(f), spring constant(K), mass for mechanical system, which can be replaced with voltage(v), inductance(L) and capacitor(1/C) respectively. Exact understanding would require knowledge of control system designing.

There are seven ideas that have shook the universe. These are:

this website is about theoretical physics and mathematics, theoretical physics, theories of phyiscs

## Science and Engineering

Science and Engineering is what we heavily dependable on now a days. Knowledge of science and mathematics can help us to understand the concepts of engineering much better. Theoretical physics is a much honorable branch of physics and science. Although theoretical physics is now an advanced branch of physics , it does not always have practical application of its theory and predictions. It relies heavily on the abstract mathematics and assumption. Modern engineering and other science have lots of similarity and they are dependent on each other. Mathematics is considered to be the foundation of all science. Logic , in similar way, is very vital to foundation of mathematics. Logic is any valid reasoning which is part of an argument itself. Logic is the rules of deduction and soundness of mathematical statements. The relation between mathematics and logic is like that between child and man. Logic is the childhood of man and man is the fatherhood of logic. They are very much alike. Much can be discussed about mathematical logic, which at the moment can be avoided as it does not give much relevance to applied science and engineering. Before going further some contradictions can be stated:
a) Let all the decimal numbers can be named in finite number of words. We can represent all these decimals as a class E. The number of such class is ℵ(0)(first transfinite ordinal). So we can arrange all such decimals as the 1st, 2 nd, 3 th, .. . Now we define a number N as follows: if nth digit in nth decimal is p , the nth digit in N is P+1 or 0(if p=9). The number N is different from all the decimals as its digit is different from that of each decimals. No matter whichever value n takes, the nth digit of N is different from nth digit of nth decimal. Neverthless we have defined N in finite number of words. So N is a member of E. Thus both N is a member and not a member of E. Seems like a contradiction!
b) Class as one can be a member of itself as many. For example, class of all classes is a class and class of terms that are not men is not men. Do all the classes that have this property form a class? If so, is it as one a member of itself as many? If it is, it is one of those classes, which as one , is not member of itself as many and vice-versa.
c) Among all the predicates, some are predicable of themselves. We take all those of which this is not the case. These are the referents(also relata) in what seems like a complex relation , namely no-predicability with identity. Now there is no predicate which attaches to all of them and no other. For this predicate is either predicable or not predictable of itself. If it is predicable of itself, it is one of the referents by relation to which it was defined. So by virtue of its definition, it is not predicable of itself. And if it is not predicable of itself, it is again one of the said referents which are predicable of themselves (by hypotheses). So it is predicable of itself. We get a contradiction. So all the predicates do not form a class.
d) That the number of all ordinal number is itself an ordinal number hints that there is no biggest ordinal number. Suppose there are N ordinal numbers. In that case all those ordinal numbers can be arranged in a series whose first term is 0 and last term is N. In that case the total number of ordinal is N+1 which contradicts the fact that total number of ordinal is N. We can not evade this paradox by saying that the series contain no last term; for in that case equally the series has an ordinal number greater than any term of the series.
This contradiction can be stated in another form: Every series which is well-ordered has an ordinal number, that series of all ordinal numbers including and up to certain ordinal number exceeds the given ordinal by one and series of all ordinal numbers is well-ordered. So it follows that series of all ordinal numbers has an ordinal number φ say, but in that case the series of all ordinal s including φ has ordinal number φ+1. So φ is not the series of all ordinals.
e) case with relation is also extraordinary. Let there be a relation R which some terms do not have to themselves. So we can consider a class w of such terms which do not have the relation R to themselves. So it is impossible that there is some term which do have the relation R to those terms of class w and no other. For if there is such a term then propositional function "x do not have the relation R with x " will be equivalent to propositional function " x has the relation R to a". Substituting a for x throughout we get a contradiction.
Suppose there is a ball(open or closed) in Euclidean 3-space. Then it possible to decompose the ball into disjoint subsets such that those subsets can be recombined to form two identicalballs similar to the original one. This paradox is known banach-tarski paradox.
Science is the blending of logic and application. Scientific study can not progress without mathematics. Mathematical logic is prior to all sciences. Science can be defined as the study of natural laws. So in this way of defining, physics comes before engineering. Applied physics can be regarded as engineering and its application. Philosophy encompasses both science and arts. So philosophy is one step ahead of science although science is more exact than philosophy. Philosophy is the most primitive knowledge. It is basically the knowledge about world and life at large. Some regard it as the intermediate knowledge between science and religion. To be specific, theology is the study of religion and science is a modern systematic study which includes technical ideas. So engineering is also a technical discipline. Scientist and engineer differ in terms of their work and responsibility. Scientists study the laws of nature and their role in the working of the universe. They do not focus on the practical aspects of these laws. On the other hand, engineers apply these laws to develop and design real world application. As an example, aeronautics engineers are concerned with making airplanes and flying them into sky using Newton's and other law of physics. So they have to rely heavily on mathematical formulas. Once they come up with appropriate formula that describes a system , they solve it to design that particular system.

Engineering as we can see, is dependent on science. The former can not progress without the later. Life will surely be troublesome without engineering applications but we could still know how the universe works as a whole since science is the study of natural laws that govern our lives and the physical world.

### Web Applications

php and mysql are now a popular web programming languages . They usually work together. HTML is hypertext markup language. So the web development involves both programming and scripting languages which includes a large class of computer languages and style sheets.

# Computer science and Engineering

Networking   |   Computerscience   |   Web development

## Elementary physics equations

All elementary physics equations include equation of work. Work is a scalar quantity which is defined by the dot product of force and distance. Work, in physics, measure of energy transfer that occurs when an object is moved over a distance by an external force at least part of which is applied in the direction of the displacement.

Impulse is a useful concept in physics. It is the force acting for small time interval δt.
These are the main equations of physics , that will occupy us here :

## Friction

Friction is the force that acts against the motion of a body or object. If there were no friction we could not move or run. Here is an example of friction which is proportional to the weight of a rigid body.

Normal force is the force acting perpendicularly to the surface where the body rests.

## Precession of a spinning top

A spinning top on the ground spins due to an external torque applied on it. But when it is left on the grounds it acts like a gyroscope. Its spinning axis precesses due to the torque produced by the gravity.

The angular momentum must be conserved. As the torque is equal to the rate of change of angular momentum ( τ = dL/dt ) the spinning axis must precess with an angular velocity that is proportional to the spinning rate of the top.

## Heisenberg's equation and Schrodinger equation

Electron's state is described by four numbers:

Schrodinger equation is derived using a wave functions representing a abstract wave.

Quantum tunneling can be explained using Schrodinger equation.

Double slit experiment reveals it all.

There was a problem with Black body radiation , which Max Planck solved by discovering his famous quantum law :

Planck reasoned that there were modes of vibration with specific energy so energy distribution is not equal . The energy of a process with frequency v is proportional to v and is equal to hv.

Quantum spin operator is used in equations of quantum mechanics and quantum field theory.

Max born interpreted Schrodinger's wave function as the probability amplitude:

Zeeman observer the effect of external magnetic on the electrons inside the atom.

## Bohr's Atomic model

Bohr's envisioned the electrons revolves in the atoms like the planets in our solar system. But this electron does not emit electromagnetic waves continuously. Electrons stay on some predefined orbits indexed by integers ( 1,2,3,. ...). When they are in an orbit , the potential energy equals the cetrifugal force.

## Classical electrodynamics

Classical electrodynamics is concerned with the moving electron in an electromagnetic field. It is not the quantized version of electromagnetism, which is called quantum electrodynamics. The Lorentz force is the force that a moving electron experiences in a electromagnetic field.

## Einstein

Einstein contributed to physics to an extent which is unimaginable. His greatest discovery was his theory of relativity although he was awarded noble prize for his photo-electric effect.

## Classical vs quantum theory

Everything in quantum physics is described by probability. Things and phenomena are fundamentally discrete in quantum physics. In quantum mechanics density matrix p determins probability of outcomes of any measurement.

Again we repeat Planck's quantum law which is different than classical law of energy of radiation.

## The legends of quantum theory

Paul dirac, Erwin Schrodinger, Max Planck are some of the legends of quantum mehcanics. Einstein was also one of them.

## Quantum entanglement

Quantum entanglement is the phenomena when two particles are produced in such a way that measurement of one can not be done independently of the other. A simple such example is the mixed state of one spin up electron and one spin down electron.

## Exact evolution of operator

The evolution of quantum mechanical operator A is the sum of commutator { A , hamiltonian H } and partial derivative A with respect to time t.

## Classical Electromagnetism

Maxwell equations are the bedrock of classical electromagnetism. These equations not only reflects a intellectual mind but also mathematical elegance. This is how it is done.

## Euler identity

We represent the two dimensional complex plance by consine as the abscissa and sine times i as the ordinate. Then it can be represented as the orthogonal 2D projection of 3D helix .

## Theory of relativity

First equation which desribes the whole matter of general relativity is the geodesic equation. It has the rather simplest form as given below:

Metric tensor is the dot product of two tangent vectors :
g(uv) = ∑ [dy`(k)/dx(v)] [dy`(k)/dx(u)]
All the special relativity formula in a single package :

One of the formula in the list says that force is time derivative of momentum. It is to be thought of time derivative of two functions mass m and velocity v. In relativity mass is a function of time. But in Newtonian mechanics mass is constant. The whole theory of relativity is concerned with mathematical functions of specific sort or other. As an example the derivation of energy-mass equivalence relation can be mentioned.

## Field equation

The equation above was developed by Einstein as part of his groundbreaking general theory of relativity in 1915. The theory revolutionized how scientists understood gravity by describing the force as a curvature of the fabric of space and time.
"It is still hilarious to me that one such mathematical equation can describe what space-time is all about," said Space Telescope Science Institute astrophysicist Mario Livio, who nominated the equation as his favorite. "All of Einstein's real genius is embodied in this equation." [Einstein Quiz: Test Your Knowledge of the Genius]
"The right-hand side of this equation describes the energy contents of our universe (including the 'dark energy' that propels the current cosmic acceleration)," Livio explained. "The left-hand side describes the geometry of space-time. The equality mirrors the fact that in Einstein's general relativity, mass and energy determine the geometry, and concomitantly the curvature, which is a manifestation of what we call gravity."
"It's a very elegant equation," said Kyle Cranmer, a physicist at New York University, adding that the equation unveils the relationship between space-time and matter and energy. "This equation tells you how they are related — how the presence of the sun warps space-time so that the Earth moves around it in orbit, etc. It also tells you how the universe evolved since the Big Bang and predicts that there should be black holes."

## Calculus

While the first two equations describe particular properties of our universe, another favorite equation can be applied to all manner of situations. The fundamental theorem of calculus forms the backbone of the mathematical method known as calculus, and links its two main ideas, the concept of the integral and the concept of the derivative.
"In simple words, [it] tells that the resultant change of a smooth and continuous quantity, such as a distance travelled, over a given time interval (i.e. the difference in the values of the quantity at the end points of the time interval) is equal to the integral of the rate of change of that quantity, i.e. the integral of the velocity," said Melkana Brakalova-Trevithick, chair of the math department at Fordham University, who chose this equation as her favorite. "The fundamental theorem of calculus (FTC) allows us to determine the net change over an interval based on the rate of change over the entire interval."
The seeds of calculus started in ancient times, but much of it was put together in the 17th century by Isaac Newton, who used calculus to describe the motions of the planets around the sun.

## Standard Model

Standard model is considered that ugliest theory in the field of theoretical physics. It is developed using the formalism of quantum field theoy. It is represented by the symmetry group SU(3)XSU(2)XU(1). The full equation can be written as :

Quantum fields in standard model lagrangian are the fermion fields, ψ, which account for "matter particles"; the electroweak boson fields W1, W2, and w3, and B; the gluon field, G(a); and the Higgs field, φ.
The Higgs field φ satisfies the Klein–Gordon equation. The photon field A satisfies the wave equation D(u)D(u)A(v) = 0.
Electroweak sector

The electroweak sector interacts with the symmetry group U(1)×SU(2)L, where the subscript L indicates coupling only to left-handed fermions.
Third line represent the Higg's mechanism which tells how the elementary particles gains mass by interaction with the Higg's field V(φ) .
The terms in the last line describe strong interaction which is mediated by gluons.
More simplified equation of the standard model will be :

## Energy of Hydrogen atom

Energy of electron in the hydrogen can be found using Scrhodinger equation. Using Dirac and Pauli equation more accurate result can be found. Yet if we apply rules of quantum electrodynamics we can get more accurate results.

## The Callan–Symanzik Equation

"The Callan-Symanzik equation is a important first-principles equation from 1970, essential for describing how naive expectations will fail in a quantum world," said theoretical physicist Matt Strassler of Rutgers University. The equation has vast range of applications, including allowing physicists to estimate the mass and size of the proton and neutron, which make up the nuclei of atoms.
Basic physics tells us that the gravitational force, and the electrical force, between two objects is commensurate to the inverse of the distance between them squared. On a simple level, the same is true for the strong nuclear force that binds protons and neutrons together to form the nuclei of atoms, and that binds quarks together to form protons and neutrons. However, tiny quantum fluctuations can slightly alter a force's dependence on distance, which has surprising consequences for the strong nuclear force.
"It prevents this force from diminishing at long distances, and causes it to trap quarks and to combine them to form the protons and neutrons of our world," Strassler said. "What the Callan-Symanzik equation does is relate this dramatic and difficult-to-calculate effect, important when [the distance] is approximately the size of a proton, to more subtle but easier-to-calculate effects that can be measured when [the distance] is much smaller than a proton."

## Minimal soap bubble equation

"The minimal surface equation somehow encrypts the beautiful soap films that form on wire boundaries when you dip them in soap filled water," said mathematician Frank Morgan of Williams College. "The fact that the equation is 'nonlinear,' involving powers and products of derivatives, is the coded mathematical suggestions for the surprising behavior of soap films. This is in contrast with more familiar linear partial differential equations, such as the heat equation, the wave equation, and the Schrödinger equation of quantum physics."

## Euler Line

This is not a topic of physics but rather geometrical corollary.

Glen Whitney, founder of the Museum of Math in New York, chose another geometrical theorem, this one having to do with the Euler line, named after 18th-century Swiss mathematician and physicist Leonhard Euler.
"Start with any triangle," Whitney expounded. "Draw the smallest circle that contains the triangle and find its center. Find the center of mass of the triangle — the point where the triangle, if cut out of a piece of paper, would be stable on a pin. Draw the three altitudes of the triangle (the lines from each corner perpendicular to the opposite side), and find the point where they all meet. The theorem or corollay is that all three of the points you just found always lie on a single straight line, known as the 'Euler line' of the triangle."
Whitney said the theorem encapsulates the beauty and power of mathematics, which often reveals surprising patterns in simple, familiar shapes.

## Fermion field

Fermion field is achieved by quantizing dirac field.

## Lagrangian of spinor (1/2) particle

Remember spinor are four component wave function. One spinor is for spin 1/2 and other spinor is for spin -1/2.

## Helmholtz equation

Helmholtz iequation is a partial differential equation describing a wave's propagation. It is also an eigenvalue problem.

## Limit

1 = 0.999999999….
This simple equation, which says that the quantity 0.999, followed by an infinite string of nines, is equivalent to one, is the favorite of mathematician Steven Strogatz of Cornell University.
"I love how simplistic it is — everyone understands what it says — yet how provocative it is," Strogatz said. "Many people don't believe it could be true. It's also beautifully balanced. The left side represents the beginning of mathematics; the right side represents the mysteries of infinity."

## Navier-Stokes equation

Navier-Stokes equation is an equation of fluid dynamics. The Navier-Stokes equations involve calculating changes in quantities like velocity and pressure. Mathematicians worry about this kind of scenario: You’re running the equations, and after some finite amount of time, they tell you a particle in the fluid is moving at infinite velocity. That would be a problem because you can’t calculate the change of an infinite value any more than you can divide by zero. Mathematicians refer to such scenarios as “blowup,” and in a blowup scenario you’d say the equations break down and solutions don’t exist.

Proving that singularity doesn’t arise (and that solutions always exist) is tantamount to proving that the maximum velocity of any particle within the fluid stays bounded below some finite number. One of the most important of these quantities is the kinetic energy in the fluid.
If you solve this equation you will get a million dollar from Clay mathematics institute.

## Fluid dynamics again

This is the equation of fluid dynamics govenring the flow of energy of the fluid and its pressure at various points.

## Newton's law of motion

These equations are known to us all along. These are famous laws of motion of Newton. Explanation is too elementary to be written down.

## How everything in solar system is being run by equations

The neutrino (μ) comes from the sun and disintegrate into other particles. Neutrino seems to obey the laws of special relativity. It seems to have lived a longer life when it enters the earth's atmosphere before disintegrating into other particles. It's normal life expectancy when at rest is very less than when it moves at a very fast speed.

## Schrodinger equation and Heisenberg's uncertainty relation

Schrodinger equation describes the evolution of wave function ψ and Heisenberg's uncertainty realtion says we can not measure the position and velocity of a particle at the same time.

## Everything in one package

All the equations are of quantum mechanics except the famous energy-mass equivalence relation of Einstein.

More equations of quantum mechanics are given below.

## Entropy equation

Not an equality but an inequality, stating that the entropy (S) of our universe always increases. Entropy can be interpreted as a measure of disorder, hence the law can be stated as the disorder of the universe increasing. An alternative view of the law is heat only flows from hot to cold objects. As well as practical uses during the industrial revolution, when designing heat and steam engines, this law also has deep consequences for our universe. It allows the definition of an arrow of time. Imagine being shown a video clip of a mug being dropped and breaking. The initial state is a mug (ordered) and the final state is a collection of pieces (disordered). You would clearly be able to tell whether the video was being played forward of backward from the flow of entropy. This would also lead on to the big bang theory, with the universe becoming hotter as you go into the past but also more ordered, leading towards the most ordered state at zeroth time; a singular point.

## Wave equation

The wave equation is a 2nd order partial differentiation equation that describes the propagation of waves. It relates the change of propagation of the wave in time to the change of propagation in space and a factor of the wave speed (v) squared. This equation is not as groundbreaking as others on this list but it is majestic and has been applied to things such as sound waves (instruments etc.), waves in fluids, light waves, quantum mechanics and general relativity.

## Feynamn Diagrams

Feynman diagrams are very simple visual representations of particle interactions. They can be appreciated superficially as a pretty image of particle physics but do not underestimate them. Theoretical physicists use these diagrams as a key tool in complicated calculations. There are rules to drawing a Feynman diagram, a particular one to note is that any particle traveling backwards in time is an antiparticle (corresponding to a standard particle but with the opposite of its electrical charge). Feynman won a noble prize for quantum electrodynamics and did lots of great work but perhaps his most well known legacy are his diagrams that every physics students learns to draw and study. Feynman even painted these diagrams all over his van.

### mechanical engineering and aerodynamics

Mechanical engineering is a sub-discipline of applied engineering. It is concerned with mechanics which was invented by Newton. That is why the name "mechanical engineering" make sense. It is the engineering used to make motor, propeller, aeroplane, centrifugal pump, heat engines and aeroplanes. The branch related to making aeroplane is known as aerodynamics. It is exclusively concerned with the dynamics of air and drag force which is the analog of friction. Fluid dynamics is another branch which is the study of fluid flow. The study of fluid dates back many centuries when Pascal developed his famous principle related to the pressure of fluid. The principle states :
The pressure of fluid at a point has infinite direction and the change of pressure on an incomprehensible fluid is transmitted throught the fluid such that same change occurs everywhere.
mechanical engineering is needed for automobile engineering. The internal combustion(I, C) engine is a kind of heat engine by which heat energy is converted to mechanical energy. I.c engine is the main component of all automobiles like car or bus. Mechanical engineering definitely has created a revolution in engineering science. Famous ship "Titanic" could not have been built without the knowledge of mechanical engineering.
Aerodynamics is copncerned with the force and drag created by air on the material body. Aeroplane is designed using the principles of aerodynamics.

### What is the fate of the universe?

The fate of the universe strongly depends on a factor of unknown value: Ω, a measure of the density of matter and energy throughout the cosmos. If Ω is greater than 1, then space-time would be "closed" like the surface of an enormous sphere. If there is no dark energy, such a universe would eventually stop expanding and would instead start contracting, eventually collapsing in on itself in an event dubbed the "Big Crunch." If the universe is closed but there is dark energy, the spherical universe would expand forever. Alternatively, if Ω is less than 1, then the geometry of space would be "open" like the surface of a saddle. In this case, its ultimate fate is the "Big Freeze" followed by the "Big Rip": first, the universe's outward acceleration would tear galaxies and stars apart, leaving all matter frigid and alone. Next, the acceleration would grow so strong that it would overwhelm the effects of the forces that hold atoms together, and everything would be wrenched apart. If Ω = 1, the universe would be flat, extending like an infinite plane in all directions. If there is no dark energy, such a planar universe would expand forever but at a continually decelerating rate, approaching a standstill. If there is dark energy, the flat universe ultimately would experience runaway expansion leading to the Big Rip. Regardless how it plays out, the universe is dying, a fact discussed in detail by astrophysicist Paul Sutter in the essay from December, 2015. Que sera, sera.

### Reference materials:

Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Quantum mechanics
Grand Design by Stephen Hawking
Higher Engineering Mathematics ( PDFDrive.com ).pdf
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