- Dirac’s equation
- Einstein’s field equation
- Maxwell’s equations
- General relativity
- Special relativity
- Schrodinger’s equation
- Uncertainty principle
- Gibb’s statistical mechanics
- Stephan-Boltzmann law
- Laplace equation
- De broglie relation-matter wave
- Navier-stokes equations
- Riemann zeta function
- Noether theorem
- Euler-lagrange equation
- Hamilton quanternion
- Standard model
- Lagrange formula
- cantor inequality
- Riemann hypothesis
- Hawking-Bekenstein entropy formula
- Heat equation
- wave equation
- poisson equation
- Wave-particle duality
- fundamental theorem of calculus
- Pythagorean theorem
- Gauss-Bonnet theorem
- universal law of gravitation
- Newton’s 2nd law of motion
- kinetic energy
- Potential energy
- 2nd law of thermodynamics
- principle of least action
- Spherical harmonics
- Cauchy residue theorem
- Callen-Symanzik equation
- Minimal surface equation
- Euler 9 point center
- Mandelbrot set
- Yang-Baxter equation
- Divergence theorem
- Baye’s theorem
- logistic map
- Einstein’s law of velocity addition
- Photoelectric effect formula
- Faraday law
- Cauchy momentum equation
- De moivreâ€™s theorem
- Fourier transform
- prime counting function
- Murphy’s law
- Summation formula
- Logarithmic spiral
- Heron’s formula
- Quadratic equation
- Euler line
- Pythagorean triple formula
- Euler’s formula
- Simplex method
- Proof of infinity of prime numbers
- Harmonic series
- Euler sums
- Cubic equation
- Quartic equation
- quintic equation
- Lorentz equation
- Euler-lagrange formula
- Euler product formula
- Euler-maclaurin formula
- Natural logarithm
- Conic sections
- exponential growth or decay
- Calculation an orbit I.e. a comet
- interesting number idea 1
- interesting number idea 2
- interesting number idea 3
- Dirac equation
Dirac equation (original)
(Î²mc^2+c(âˆ‘ of (1 to 3) for n=Î±npn))Ïˆ(x,t)=iâ„âˆ‚Ïˆ(x,t)/âˆ‚t
The Dirac differential equation from quantum mechanics was formulated in 1928 which predicted the existence of antimatter, which are particle of the same mass and spin, but have an opposite charges than their counterparts of matter.
2. Einstein field equation
The einstein field equations, or the einstein-hilbert equations is used to describe gravity in a classical way. It uses geometry to model gravity’s effects.
3. Maxwell’s equations
James clerk maxwell formulated 4 differential equations to describe how charged particles produce an electric and magnetic force. They calculate the motion of particles in electric and magnetic fields. They describe how electric charges and electric currents create electric and magnetic fields, and vice versa.
The 1st equation is used to calculate the electric field produced by a charge.
The 2nd equation is used to calculate the magnetic field.Â The 3rd equation, ampere’s law, shows how the magnetic fields circulate around electric currents and time varying electric fields. The 4th equation. Faradayâ€™s law, shows how the electric fields circulate around time varying magnetic fields.
4.Â General relativityâ€”
Albert einstein, in 1915, formed the general theory of relativity which deals with space and time, two aspects of spacetime. Spacetime curves when there is gravity, matter, energy, and momentum. Central the the general theory of relativity is the principle of equivalence. The theory shows that light curves in an accelerating frame of reference. It also asserts that light will bend and it will slow down in the presence of a massive amount matter.
5. SPECIAL RELATIVITY
The Lorentz Transformations (the mathematical basis for the special theory of relativity-
yâ€²=y, and zâ€²=z.
an object’s mass, total energy and momentum are related by the equation-
p=0 for an object at rest, so the equation becomes
An object at rest still has energy, called its rest mass, which is-
The special theory of relativity asserts that the speed of light is the same no matter what speed the observer travels. It also explains what is relative and what is absolute about time, space, and motion. It further describes how mass increases, length shrinks, time slows down for objects moving close to the speed of light, and that a person traveling close to the speed of light would age less than would a stationary person.
6. Schrodinger equation
(For one particle that only moves in one direction in space)
where i is the square root of -1, â„ is the reduced Planck’s constant, t is time, x is a position,Î¨(x,t) is the wave function, and V(x) is the potential energy, an as yet not chosen function of position. The left hand side is equivalent to the hamiltonian energy operator acting on Î¨.
(where the first equation is solely dependent on time)
T(t), and the second equation depends only on position Ïˆ(x), and where E is just a number. The first equation can be solved immediately to give T(t)=e^(âˆ’i(Et))â„
where e is ruler’s number.
the function of two variables can be written as the product of two different functions of a single variable: Î¨(x,t)=Ïˆ(x)T(t) then the wave equation can be rewritten as two distinct differential equations iâ„dT(t)dt=ET(t)
This is a differential equation that is the basis of quantum mechanics. It is one of the most precise theories of how subatomic particles behave as fully as possible. This equation defines a wave function of a particle or group of particles that have a certain value at every point in space for every given time. the wave function contains all information that can be known about a particle or system. The wave function gives real values relating to physical properties such as position, momentum, energy, etc.
7. Uncertainty principle
The range of error in position (x) times the range of error in momentum (p) is about equal to or greater than the dirac constant planks constant divided by 4pi.
This principle says that trying to pin a thing down to one definite position will make its momentum less well pinned down, and vice-versa.
8. GIBB’S STATISTICAL MECHANICS
canonical partition function
grand partition function
Helmholtz free energy
Statistical mechanics is a branch of theoretical physics which uses probability theory to study the average behavior of a
mechanical system, where the state of the system is uncertain.
Statistical mechanics is commonly used to explain the thermodynamic
behavior of large systems.
9. Stefan Boltzmann law
where Ïƒ is the Stefan-Boltzmann constant, which is equal to 5.670 373(21) x 10-8 W m-2 K-4, and where R is the energy radiated per unit surface area and per unit time. T is temperature, which is measured in kelvin scale. this law is only usable for the energy radiated by blackbodies but is still useful none the less.
In quantum physics, the Stefan-Boltzmann law (sometimes called Stefan’s Law) states that the black body radiation energy emitted by an object is directly proportional to the temperature of the object raised to the fourth power.
10. Mass energy equivalence
In physics, mass energy equivalence asserts that anything having mass has an equivalent amount of energy and vice versa. these fundamental quantities are directly related to one another.
This formula states that the equivalent energy (E) can be calculated as the mass (m) multiplied by the speed of light (c = about 3Ã—108 m/s) squared. Similarly, anything having energy exhibits a corresponding mass m given by its energy E divided by the speed of light squared cÂ².
11. Laplace’s equation
âˆ‡2Ï†=0 or Î”Ï†=0 where âˆ†Â =Â âˆ‡2 is the laplace operator and
Ï† is a scalar function.
In mathematics, Laplace’s equation is a second-order partial differential equation. The solutions of Laplace’s equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials. In the study of heat conduction, the Laplace equation is the steady-state heat equation.
12. DE BROGLIE RELATION/Matter wave
Where Î» is the wavelength of the object, h is Planck’s constant, m is the mass of the object, and v is the velocity of the object. An alternate but correct version of this formula is
Where p is the momentum. (Momentum is equal to mass times velocity). These equations merely say that matter exhibits a particle-like nature in some circumstances, and a wave-like characteristic at other times.
13. Navier Stokes equations
The Navier Stokes equations describe the motion of fluids. The equations result from applying newton’s 2nd law to fluid dynamics with the belief that the fluid stress is the sum of a diffusing vicious term (in relation to the gradient of velocity), plus a pressure term. They are very useful because they describe the physics of many things. They may be used to model weather, ocean currents, water flow in a pipe, the air’s flow around a wing, and the motion of stars inside a galaxy. The Navier Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Together with Maxwell’s equations they can be used to model and study magnetohydrodynamics. The Navier Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (existence), or that if they do exist, then they do not contain any singularities (or infinity or discontinuity) (smoothness). These are called the navier-stokes existence and smoothness problems. The Navier Stokes equations dictate not position but rather velocity. A solution of the Navier Stokes equations is called a velocity field or flow field, which is a description of the velocity of the fluid at a given point in space and time. Once the velocity field is solved for, other quantities of interest (such as flow rate or drag force) may be found. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, however for visualization purposes one can compute various trajectories.
14. Riemann zeta function
Î¶(s)=âˆ‘n=1 to âˆž 1ns, Re(s)>1.
Re(s) is the real part of the complex number
s. For example, if s=a+ib, then Re(s)=a. (where i^2=âˆ’1)
Riemann zeta function Î¶(s) in the complex plane. The color of a point s shows the value of Î¶(s): strong colors are for values close to zero and hue encodes the value’s argument. The white spot at s= 1 is the pole of the zeta function; the black spots on the negative real axis and on the critical line Re(s) = 1/2 are its zeros.
The coloring of the complex function-values used above: positive real values are presented in red.
In mathematics, the Riemann zeta function, is a prominent function of great significance in number theory. It is so important because of its relation to the distribution of prime numbers. It also has applications in other areas such as physics, probability theory, and applied statistics.
The riemann hypothesis is a conjecture about the distribution of the zeros of the Riemann zeta function. Many mathematicians consider the Riemann hypothesis to be the most important unsolved problem in pure mathematics.
15. Noether’s theorem
Emmy noether was an influential mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics.
Noether’s theorem can be stated informally
If a system has a continuous symmetry property, then there are corresponding quantities whose values are conserved in time.
A more sophisticated version of the theorem involving fields states that:
To every differentiable symmetry generated by local actions, there corresponds a conserved current.
16. Euler Lagrange equation
In the calculus of variations, the Euler Lagrange equation, Euler’s equation, or Lagrange’s equation (although the latter name is ambiguous), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.
Because a differentiable functional is stationary at its local maxima and minima, the Euler Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to format’s theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero. In Lagrangian mechanics, because of Hamilton’s principle of stationary action, the evolution of a physical system is described by the solutions to the Euler Lagrange equation for the action of the system. In \classical mechanics, it is equivalent to newton’s law of motion, but it has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.
a + bi + cj + dk
where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.
In mathematics, the quaternions are a number system that extends the complex numbers. they are applied to in 3-dimensional space. A feature of quaternions is that multiplication of two quaternions is non-commutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space] or equivalently as the quotient of two vectors.
Quaternions find uses in both theoretical and applied mathematics, in particular for calculations involving 3-dimensional rotations such as in 3-dimensional computer graphics, computer vision and crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as euler angles and rotation matrices, or as an alternative to them, depending on the application.
18. Standard Model (mathematical formulation)
|SÂ =Â â„||âŽ¡
|S(0)Â =Â â„||âŽ¡
|S(Â½)Â =Â â„||âŽ¡
|S(1)Â =Â â„||âŽ¡
|S(1Â½)Â =Â â„||âŽ¡
|S(2)Â =Â â„||âŽ¡
Elementary particles have an intrinsic spin angular momentum S. The adjective intrinsic means innate or essential to the thing itself. Elementary particles don’t have spin because someone is spinning them. They just spin or rather, they just have a measurable quantity with the same units as angular momentum. In current physics, elementary particles are featureless like a mathematical point. In order for something to be perceived as spinning, the thing spinning would need something like a “front” and a “back”. Featureless, point particles don’t have anything like that. Particle physics is best described with mathematics. Spin is a convenient label for a measurable quality and not a description of reality.
Every elementary particle has associated with it a spin quantum number s (often called the spin number or just the spin), where s is any whole number multiple of a half. Fermions have half integral spin quantum numbers (Â½, 1Â½, 2Â½, etc.) and bosons have integral spin quantum numbers (0, 1, 2, etc.). No spin numbers are possible in between these. Spin is a quantized quantity.
The elementary fermions have a spin Â½. Particles made from combinations of fermions will have an overall spin that’s a combination of the individual spins. A baryon composed three quarks will combine to an overall spin of Â½ or 1Â½, since those are the only possible, non-negative combinations of Â½Â Â±Â Â½Â Â±Â Â½. That shows that all baryons (like protons and neutrons, for example) are also fermions. Likewise, a meson composed of a quark and an antiquark will combine to an overall spin of 0 or 1 since those are the only possible, non-negative combinations of Â½Â Â±Â Â½. That shows that all all mesons (like the pion of the residual strong interaction, for example) are also bosons.
The force carrying bosons of the standard model (gluons, photons, and the W and Z) have spin 1 since they go with vector fields. The Higgs boson corresponds to a scalar field so it has spin 0. If the particle of the gravitational field is ever discovered, it would be called a graviton and would have a spin 2 since it corresponds to a tensor field. A tensor is a mathematical object that’s more complex than a vector, which is in turn more complex than a scalar. See the trend? A scalar field with no direction gets a particle with spin 0. A vector field with a direction gets a particle with spin 1. A tensor field that stretches and squeezes space in two directions gets a particle with spin 2.
All fundamental and composite particles have a spin quantum number s (lowercase). This is associated with a spin angular momentum S (uppercase). The SI unit of angular momentum is the kilogram meter squared per second [kgm2/s] or, equivalently, the joule second [Js], which is much too large for elementary particles. Instead â„ (h bar), also known as the reduced planck constant (â„Â =Â h/2Ï€), is used.Â the spin quantum number s (which is just a number) and the spin angular momentum S (which is a number with a unit) are not numerically the same. Instead, they are related by a non-obvious equation.
19. LAGRANGE FORMULA
Lagrange’s equations (First kind)
Lagrange’s equations (Second kind)
Lagrangian mechanics is a reformulation of classical mechanics.
In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.
20. CANTOR’S INEQUALITY/Cantor’s theorem
In elementary set theory, Cantor’s theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power sets of A, ð’«(A)) has a strictly greater cardinality than A itself. For finite sets, Cantor’s theorem can be seen to be true by simple enumeration of the number of subsets. Counting the empty subset, a set with n members has 2n subsets, so that if card(A) = n, then card(ð’«(A)) = 2n, and the theorem holds because 2n > n is true for all non-negative integers.
Much more significant is Cantor’s discovery of an argument that is applicable to any set, which showed that the theorem holds for infinite sets, countable or uncountable, as well as finite ones. As a particularly important consequence, the power set of the set of natural numbers, a countably infinite set with cardinality â„µ0 = card(â„•), is uncountably infinite and has the same size as the set of real numbers, a cardinality often referred to as theÂ cardinality of the continuum:ð” = card(â„) = card(ð’«(â„•)). The relationship between these cardinal numbers is often expressed symbolically by the equality
the theorem implies that there is no largest cardinal number (colloquially, “there’s no largest infinity”
21. Riemann hypothesis
The Riemann hypothesis is a mathematical conjecture. Many people think that finding a proof of the hypothesis is one of the hardest and most important unsolved problems of pure mathematics.
The hypothesis is named after Bernhard riemann. It is about a special function, the riemann zeta function. This function inputs and outputs complex numbers values. The inputs that give the output zero are called zeros of the zeta function. Many zeros have been found. The “obvious” ones to find are the negative even integers. This follows from Riemann’s functional equation. More have been computed and have real part 1/2. The hypothesis states all the undiscovered zeros must have real part 1/2.
The functional equation also says all zeros (except the “obvious” ones) must be in the critical strip: real part is between 0 and 1. The Riemann hypothesis says more: they are on the line given, in the image on the right (the white dots). If the hypothesis is false, this would mean that there are white dots which are not on the line given.
If proven correct, this would allow mathematicians to better describe how the prime numbers are placed among whole numbers. The Riemann hypothesis is so important, and so difficult to prove, that the Clay Mathematics Institute has offered $1,000,000 to the first person to prove it.
22. HAWKING-BEKENSTEIN ENTROPY FORMULA
In physics, black-hole thermodynamics is the area of study that seeks to reconcile the laws of thermodynamics with the existence of Black-hole event horizons. As the study of the statistical mechanics of black-body radiation led to the advent of the theory of quantum mechanics, the effort to understand the statistical mechanics of black holes has had a deep impact upon the understanding of quantum gravity, leading to the formulation of the holographic principle.
The 2nd law of thermodynamics requires that black holes have entropy. If black holes carried no entropy, it would be possible to violate the second law by throwing mass into the black hole. The increase of the entropy of the black hole more than compensates for the decrease of the entropy carried by the object that was swallowed.
Starting from theorems proved by stephen hawkings,Â jacob bekenstein conjectured that the black-hole entropy was proportional to the area of its event horizon divided by the Planck area. In 1973 Bekenstein suggested
as the constant of proportionality, asserting that if the constant was not exactly this, it must be very close to it. The next year, in 1974, Hawking showed that black holes emit thermal hawking radiation corresponding to a certain temperature (Hawking temperature)
Using the thermodynamics relationship between energy, temperature and entropy, Hawking was able to confirm Bekenstein’s conjecture and fix the constant of proportionality at
A is the area of the event horizon, calculated at 4Ï€R^2,k is Boltzmann’s constant, and â„“P=Gâ„/c^3
is the planck length. This is often referred to as the Bekensteinâ Hawking formula. The subscript BH either stands for “black hole” or “Bekensteinâ Hawking”. The black-hole entropy is proportional to the area of its event horizon A.
The laws of black-hole mechanics
The four laws of black-hole mechanics are physical properties that black holes are believed to satisfy. The laws, analogous to the laws of thermodynamics, were discovered by Brandon carter, stephen hawking, and James Bardeen.
Statement of the laws
The laws of black-hole mechanics are expressed in geometrized units.
The zeroth law
The horizon has constant surface gravity for a stationary black hole.
The first law
For perturbations of stationary black holes, the change of energy is related to change of area, angular momentum, and electric charge by
E is the energy, Îº is the surface gravity, A is the horizon area Î©
The second law
The horizon area is, assuming the weak energy condition, a non-decreasing function of time: dA/dtâ‰¥0.
This “law” was superseded by Hawking’s discovery that black holes radiate, which causes both the black hole’s mass and the area of its horizon to decrease over time.
The third law
It is not possible to form a black hole with vanishing surface gravity.
Îº=0 is not possible to achieve.
23. HEAT EQUATION
For a function u(x,y,z,t) of three spatial variables (x,y,z) and the time variable t, the heat equation is
More generally in any coordinate system:
The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time. where Î± is a positive constant, and Î” or âˆ‡2 denotes the laplace operator. In the physical problem of temperature variation, u(x,y,z,t) is the temperature and Î± is the Thermal diffusivity. For the mathematical treatment it is sufficient to consider the case Î±Â =Â 1.
In the heat equation in two dimensions predicts that if one area of an otherwise cool metal plate has been heated, say with a torch, over time the temperature of that area will gradually decrease, starting at the edge and moving inward. Meanwhile the part of the plate outside that region will be getting warmer. Eventually the entire plate will reach a uniform intermediate temperature.
The heat equation is of fundamental importance in diverse scientific fields. In mathematics, it is the prototypical parabolic partial differential equation. In probability theory, the heat equation is connected with the study of brownian motion via the Fokker-planck equation.In financial mathematics, it is used to solve the black-scholes partial differential equation. The diffusion equation, a more general version of the heat equation, arises in connection with the study of chemical diffusion and other related processes.
The heat equation is used in probability and describes random walks. It is also applied in financial mathematics for this reason.
It is also important in riemannian geometry and thus topology: it was adapted by richard s. Hamilton when he defined the Ricci flow that was later used by Grigori perelmanto solve the topological poincare conjecture.
24. Wave equation
where i is the imaginary number, Ïˆ (x,t) is the wave function, Ä§ is the reduced planck constant, t is time, x is position in space, Ä¤ is a mathematical object known as the Hamilton operator. The reader will note that the symbol âˆ‚/âˆ‚t denotes that the partial derivative of the wave function is being taken.
Equations that describe waves as they occur in nature are called wave equations. Waves as they occur in rivers, lakes, and oceans are similar to those of sound and light. The problem of having to describe waves arises in fields like acoustics, electromagnetic, and fluid dynamics.
Historically, the problem of a vibrating string such as that of a musical instruments was studied.Â In 1746, d’Alambert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.
In quantum mechanics, the Wave function, usually represented by Î¨, or Ïˆ, describes the probability of finding an electron somewhere in its matter wave. To be more precise, the square of the wave function gives the probability of finding the location of the electron in the given area, since the normal answer for the wave function is usually a complex number. The wave function concept was first introduced in the legendary schrodinger equation.
25. Poisson’s equation
When f=0 identically we obtain laplace’s equation.
In mathematics, Poisson’s equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of laplace’s equation, which is also frequently seen in physics. Poisson’s equation may be solved using a green’s function.
26.Wave particle duality
Wave particle duality is perhaps one of the most confusing concepts in physics, because it is so unlike anything we see in the ordinary world.
Physicists who studied light in the 1700s and 1800s were having a big argument about whether light was made of particles shooting around like tiny bullets, or waves washing around like water waves. Light seems to do both. At times, light seems to go only in a straight line, as if it were made of particles. But other experiments show that light has a frequency and wavelength, just like a sound wave or water wave. Until the 20th century, most physicists thought that light was either one or the other, and that the scientists on the other side of the argument were simply wrong.
Wave particle duality means that all particles show both wave and particle properties. This is a central concept of quantum mechanics. Classical concepts like “particle” and “wave” do not fully describe the behavior of quantum-scale objects.
An electron has a wavelength called the “de Broglie wavelength”. It can be calculated using the equation
Î»D is the de Broglie wavelength.h is Planck’s constant Ï
is the momentum of the particle.
This made the idea that electrons in atoms show a standing wave pattern.
Waves as particles
The photoelectric effect shows that a light photon which has enough energy (a high enough frequency), can cause an electron to be released off a metal’s surface. Electrons in this case can be called photoelectrons.
27. Fundamental theorem of calculus
The fundamental theorem of calculus is central to the study of calculus. It is the theorem that shows the relationship between the derivative and the integral and between the definite integral and the indefinite integral. It is broken into two parts, the first fundamental theorem of calculus and the second fundamental theorem of calculus.
The first fundamental theorem of calculus states that if the function f is continuous, then
This means that the derivative of the integral of a function f with respect to the variable t over the interval [a,x] is equal to the function f with respect to x. This describes the derivative and integral as inverse processes.
Second Fundamental Theorem of Calculus
The second fundamental theorem of calculus states that if the function f(x) is continuous, then
âˆ«a to b f(x)dx=F(b)âˆ’F(a)
This means that the definite integral over an interval [a,b] is equal to the antiderivative evaluated at b minus the antiderivative evaluated at a. This gives the relationship between the definite integral and the indefinite integral (antiderivative).
28. Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras’s theorem is a statement about the sides of a right triangle.
One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side.
The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. If the lengths of the legs are a and b, and the length of the hypotenuse is c, then,
Pythagorean Triples or Triplets are three whole numbers which fit the equation
The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then
9+16=25. This can also be shown as 3^2+4^2=5.
The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because
29. Gauss Bonnet theorem
The Gauss Bonnet theorem or Gauss Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry (in the sense of curvature) to their topology (in the sense of the euler characteristic).
where dA is the element of area of the surface, and ds is the line element along the boundary of M. Here,Ï‡(M) is the Euler characteristic of M.
M. If the boundary âˆ‚M is piecewise smooth, then we interpret the integralâˆ«âˆ‚Mkgds as the sum of the corresponding integrals along the smooth portions of the boundary, plus the sum of the angles by which the smooth portions turn at the corners of the boundary.
30. Newton’s law of universal gravitation
Newton’s universal law of gravitation is a physical law that describes the attraction between two objects with mass.
In this equation:
â€¢ Fg is the total gravitational force between the two objects.
â€¢ G is the gravitational constant.
â€¢ m1 is the mass of the first object.
â€¢ m2 is the mass of the second object.
â€¢ r is the distance between the centres of the objects.
In SI units, Fg is measured in newtons (N), m1 and m2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6.674Ã—10âˆ’11Â N m2 kg
31. Newton’s 2nd law of motion
â€¢For a particle of mass m, the net force F on the particle is equal to the mass m times the particle’s acceleration a.
32. Kinetic energy
Kinetic energy is the energy that an object has because of its motion. This energy can be converted into other kinds, such as gravitational or electric potential energy, which is the energy that an object has because of its position in a gravitational or electric field.
Translational kinetic energy
The translational kinetic energy of an object is:
where m is the mass (resistance to linear acceleration or deceleration); v
is the linear velocity.
Rotational kinetic energy
The rotational kinetic energy of an object is:
where I is the moment of inertia (resistance to angular acceleration or deceleration, equal to the product of the mass and the square of its perpendicular distance from the axis of rotation);
Ï is the angular velocity.
33. Potential energy
Potential energy is the energy that an object has because of its position on a gradient of potential energy called a potential field.
Actual energy (E = hf) is nonzero frequency angular momentum.
Potential energy (rest mass) is zero frequency angular momentum.
The potential fields are irrotationally radial (“electric”) fluxes of the vacuum and divide into two classes:
The gravitoelectric fields;
The electric fields.
The potential energy is negative. It is not a mere convention but a consequence of conservation of energy in the zero-energy universe as an object descends into a potential field, its potential energy becomes more negative, while its actual energy becomes more positive, and, in accordance with the 2nd law of thermodynamics, tends to be radiated away, so that the object acquires a net negative potential energy, also known as the object’s binding energy.
In accordance with the minimal total potential energy principle, the universe’s matter flows towards ever more negative total potential energy. This cosmic flow is time.
Gravitational potential energy
Self gravitating sphere
The gravitational potential energy of a massive spherical cloud is proportional to its radius and causes the sphere to fall towards its own centre.
Initially, the cold and rarefied cloud is transparent to its own blackbody radiation, so the collapse is isothermal (because any increase in temperature is immediately radiated away) and continues on a free-fall timescale tff, which is inversely proportional to the square root of the sphere’s density Ï:tff=(3Ï€32GÏ)1/2.
If an object is lifted a certain distance from the surface from the earth, the force experienced is caused by weight and height. Work is defined as force over a distance, and work is another word for energy. This means gravitational potential energy is equal to
is the force of gravity Î”h is the change in height or U=mgh
Total work done by gravitational potential energy in a moving object from position 1 to position 2 can be found by:
Î”W=U1âˆ’U2 or Î”W=mgh1âˆ’mgh2
where m is the mass of the object g is the acceleration caused by gravity (constant) h1 is the first position h2
is the second position
Electric potential energy
Electric potential energy is experienced by charges both different and alike, as they repel or attract each other. Charges can either be positive (+) or negative (-), where opposite charges attract and similar charges repel. If two charges were placed a certain distance away from each other, the potential energy stored between the charges can be calculated by:
where k is 1/4Ï€Ñ” (for air or vacuum it is 9x109Nm^2/C^2)Q is the first charge q is the second charge r is the distance apart.
Elastic potential energy
Elastic potential energy is experienced when a rubbery material is pulled away or pushed together. The amount of potential energy the material has depends on the distance pulled or pushed. The longer the distance pushed, the greater the elastic potential energy the material has. If a material is pulled or pushed, the potential energy can be calculated by:
U =12kx^2 where k is the spring force constant (how well the material stretches or compresses) x is the distance the material moved from its original position.
34. Second law of thermodynamics
The second law of thermodynamics says that when energy changes from one form to another form, or matter moves freely, entropy (disorder) increases, in a closed system.
Differences in temperature, pressure, and density tend to even out horizontally after a while. Due to the force of gravity, density and pressure do not even out vertically. Density and pressure on the bottom will be more than at the top.
Entropy is a measure of spread of matter and energy to everywhere they have access.
The most common wording for the second law of thermodynamics is essentially due to Rudolf Clausius:
It is impossible to construct a device which produces no other effect than transfer of heat from lower temperature body to higher temperature body
In other words, everything tries to maintain the same temperature over time.
There are many statements of the second law which use different terms, but are all equal. Another statement by Clausius is:
Heat cannot of itself pass from a colder to a hotter body.
An equivalent statement by Lord Kelvin is:
A transformation whose only final result is to convert heat, extracted from a source at constant temperature, into work, is impossible.
The second law only applies to large systems. The second law is about the likely behavior of a system where no energy or matter gets in or out. The bigger the system is, the more likely the second law will be true.
In a general sense, the second law says that temperature differences between systems in contact with each other tend to even out and that work can be obtained from these non-equilibrium differences, but that loss of thermal energy occurs, when work is done and entropy increases. Pressure, density and temperature differences in an isolated system, all tend to equalize if given the opportunity; density and pressure, but not temperature, are affected by gravity. A heat engine is a mechanical device that provides useful work from the difference in temperature of two bodies.
The law that entropy always increases, holds, I think, the supreme position among the laws of nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwellâ€™s equations â€” then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation â€” well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can give you no hope; there is nothing for it but to collapse in deepest humiliation.
–Sir Arthur Stanley Eddington, The Nature of the Physical World (1927)
The tendency for entropy to increase in isolated systems is expressed in the second law of thermodynamics — perhaps the most pessimistic and amoral formulation in all human thought.
There are almost as many formulations of the second law as there have been discussions of it.
–Philosopher / Physicist P.W. Bridgman, (1941)
35. Principle of least actionâ€”
The principle of least action â€“ or, more accurately, the principle of stationary action â€“ is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive newtonian, lagrangian and hamiltonian equations of motion, and even general relativity. The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in morse theory.Â maupertuisâ€™ principle and Hamiltonâ€™s principle exemplify the principle of stationary action.
36. SPHERICAL HARMONICSâ€”
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions (sines and cosines) are used to represent functions on a circle via fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
37. Cauchy Residue theoremâ€”
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy’s residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals as well. It generalizes the cauchy integral theorem and cauchy integral formula. From a geometrical perspective, it is a special case of the generalized stokeâ€™s theorem.
38. Callanâ€“Symanzik equationâ€”
Î²(g) being the beta function and Î³ the scaling of the fields.
In quantum electrodynamics, this equation takes the form
n and m being the number of electrons and photons respectively.
In physics, the Callanâ€“Symanzik equation is a differential equation describing the evolution of the n-point correlation functions under variation of the energy scale at which the theory is defined and involves the beta-function of the theory and the anomalous dimensions.
39. MINIMAL SURFACE EQUATIONâ€”
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature. The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas.
costaâ€™s minimal surface
Classical examples of minimal surfaces include:
â€¢the plane, which is a trivial case
â€¢ Â Â catenoids: minimal surfaces made by rotating a catenary once around its directrix
â€¢ Â Â helicoids: A surface swept out by a line rotating with uniform velocity around an axis perpendicular to the line and simultaneously moving along the axis with uniform velocity
Surfaces from the 19th century golden age include:
â€¢.Â Â Schwarz minimal surfaces: triply periodic surfaces that fill R3
â€¢riemannâ€™s minimal surface: A posthumously described periodic surface
â€¢the enneper surface
â€¢the henneberg surface: the first non-orientable minimal surface
â€¢bourâ€™s minimal surface
Modern surfaces include:
â€¢the gyroid: One of Schoen’s 1970 surfaces, a triply periodic surface of particular interest for liquid crystal structure
â€¢the saddle tower family: generalizations of scherzoâ€™s 2nd surface
â€¢costaâ€™s minimal surface: Famous conjecture disproof. This was extended to produce a family of surfaces with different rotational symmetries.
â€¢the chen-gackstatter surface family, adding handles to the Enneper surface.
40. EULERâ€™S 9 POINT CENTER/Nine-point centerâ€”
In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle. It is so-called because it is the center of the 9-point circle, a circle that passes through nine significant points of the triangle: the midpoints of the three edges, the feet of the three altitudes, and the points halfway between the orthocenter and each of the three vertices.
41 Â MANDELBROT SETSâ€”
The Mandelbrot set is a famous example of a fractals in mathematics.The Mandelbrot set is important for the chaos theory. The edging of the set shows a self-similarity, which is not perfect because it has deformations.
The Mandelbrot set can be explained with the equation zn+1 = zn2 + c. In that equation, c and z are complex numbers and n is zero or a positive integer (natural numbers). Starting with z0=0, c is in the Mandelbrot set if the absolute value of zn never becomes larger than a certain number (that number depends on c), no matter how large n gets. things typically considered to be “rough”, a “mess” or “chaotic”, like clouds or shorelines, actually had a “degree of order”. The equation zn+1 = zn2 + c was known long before Benoit Mandelbrot used a computer to visualize it.
Images are created by applying the equation to each pixel in an iterative process, using the pixel’s position in the image for the number ‘c’. ‘c’ is obtained by mapping the position of the pixel in the image relative to the position of the point on the complex plane. The shape of the Mandelbrot Set is represented in black in the image on this page. For example, if c = 1 then the sequence is 0, 1, 2, 5, 26,â€¦, which goes to infinity. Therefore, 1 is not an element of the Mandelbrot set, and thus is not coloured black.
On the other hand, if c is equal to the square root of -1, also known as i, then the sequence is 0, i, (âˆ’1 + i), âˆ’i, (âˆ’1 + i), âˆ’iâ€¦, which does not go to infinity and so it belongs to the Mandelbrot set. When graphed to show the entire Set, the resultant image is striking, pretty, and quite recognizable.
There are many variations of the Mandelbrot set, such as Multibrot, Buddhabrot, and Nebulabrot. Multibrot is a generalization that allows any exponent: zn+1 = znd + c. These sets are called multibrot sets. The Multibrot set for d = 2 is the Mandelbrot set.
42. Yangâ€“Baxter equationâ€”
In physics, the Yangâ€“Baxter equation (or star-triangle relation) is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that in some scattering situations, particles may preserve their momentum while changing their quantum internal states. It states that a matrix
R, acting on two out of three objects, satisfies
In one dimensional quantum systems,
R is the scattering matrix and if it satisfies the Yangâ€“Baxter equation then the system is integrable. The Yangâ€“Baxter equation also shows up when discussing knot theory and the braid groups where
R corresponds to swapping two strands. Since one can swap three strands two different ways, the Yangâ€“Baxter equation enforces that both paths are the same.
43. DIVERGENCE THEOREMâ€”
In vector calculus, the divergence theorem, also known as Gauss’s theorem or Ostrogradsky’s theorem, is a result that relates the flow (that is, flux) of a vector field through a surface to the behavior of the vector field inside the surface. More precisely, the divergence theorem states that the outward flux of a vector field through a closed surface is equal to the volume integral of the divergence over the region inside the surface. Intuitively, it states that the sum of all sources (with sinks regarded as negative sources) gives the net flux out of a region. The divergence theorem is an important result for the mathematics of physics and engineering, in particular in electrostatics and fluid dynamics.
In physics and engineering, the divergence theorem is usually applied in three dimensions. However, it generalizes to any number of dimensions. In one dimension, it is equivalent to the fundamental theorem of calculus. In two dimensions, it is equivalent to greenâ€™s theorem. The theorem is a special case of the more general stokeâ€™s theorem.
44. Bayes’ theoremâ€”
In probability theory and applications, Bayes’ theorem shows the relation between a conditional probability and its reverse form. For example, the probability of a hypothesis given some observed pieces of evidence and the probability of that evidence given the hypothesis.
45. Logistic mapâ€”
is a number between zero and one that represents the ratio of existing population to the maximum possible population.
46. EINSTEINâ€™S LAW OF VELOCITY ADDITION/Velocity-addition formulaâ€”
In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive lorentz transformations, so they also relate different frames. Accompanying velocity addition is a kinematic effect known as thomas procession, whereby successive non-collinear Lorentz boosts become equivalent to the composition of a rotation of the coordinate system and a boost.
Standard applications of velocity-addition formulas include the doppler shift, doppler navigation, the aberration of light, and the dragging of light in moving water. It was observed by galilei that a person on a uniformly moving ship has the impression of being at rest and sees a heavy body falling vertically downward. This observation is now regarded as the first clear statement of the principle of mechanical relativity. The cosmos of Galileo consists of absolute space and time and the addition of velocities corresponds to composition of galilean transformations. The relativity principle is called galilean relativity. It is obeyed by newtonian mechanics.
According to the theory of special relativity, the frame of the ship has a different clock rate and distance measure, and the notion of simultaneity in the direction of motion is altered, so the addition law for velocities is changed. The cosmos of special relativity consists of Minkowski spacetime and the addition of velocities corresponds to composition of lorentz transformations. In the special theory of relativity Newtonian mechanics is modified into relativistic mechanics.
47. PHOTOELECTRIC EFFECT FORMULAâ€”
The photoelectric equation involves; h = the Planck constant 6.63 x 10-34 J s. f = the frequency of the incident light in hertz (Hz) … Ek = the maximum kinetic energy of the emitted electrons in joules (J)
The photoelectric effect is the emission of electrons or other free carriers when light is shone onto a material. Electrons emitted in this manner can be called photo electrons. The phenomenon is commonly studied in electronic physics, as well as in fields of chemistry, such as quatuum chemistry or electrochemistry.
48. Faraday’s law of inductionâ€”
Faraday’s law of induction is one of the basic laws of electromagnetism. The law explains the operation principles of generators, transformers and electric motors.
Faraday’s law of induction says that when a magnetic field changes, it causes a voltage. To describe the law, the magnetic flux and also a surface with a wire loop as border. This leads to the following surface integral:
â€¢Î¦B is the magnetic flux
â€¢ dA is a small part of the moving surface
â€¢ B is the magnetic field
When the flux changes, it produces electromotive force. The flux changes when B changes or when the wire loop is moved or deformed, or when both happens. The electromotive force can then be calculated with the following equation:
is the electromotive force
â€¢ N is the number of loops the wire makes
â€¢Î¦B is the magnetic flux of one loop
49. Cauchy momentum equationâ€”
The Cauchy momentum equation is a vector partial differential equation put forth by cauchy that describes the non-relativistic momentum transport in any continuum. In convective (or Lagrangian) form it is written:
where Ï is the density at the point considered in the continuum (for which the continuity equation holds), Ïƒ is the stress tensor, and g contains all of the body forces per unit mass (often simply gravitational acceleration). u is the flow velocity vector field, which depends on time and space.
Notably, it can be written, through an appropriate change of variables, also in conservation (or Eulerian) form:
where j is the momentum density at the point considered in the continuum (for which the continuity equation holds), F is the flux associated to the momentum density, and s contains all of the body forces per unit volume.
50. De Moivre’s formulaâ€”
In mathematics, de Moivre’s formula, for any complex number z and integer n,
|z| is the modulus of z, and arg z is the argument of z. Here, e
is rulerâ€™s number, with |z|e^i,argz often being called the polar form of a complex number. The formula connects complex numbers and trigonometry.
51. Fourier transformâ€”
The Fourier transform of a function
is given by
F(Î±)=âˆ«âˆ’âˆž to +âˆžf(x)eâˆ’2Ï€iÎ±xdx
The inverse Fourier transform is given by
f(x)=âˆ«âˆ’âˆž to +âˆžF(Î±)e+2Ï€ixÎ±dÎ±
The Fourier transform is a mathematical function that can be used to show the different parts of a continuous signal. It is most used to convert from time domain to frequency domain. Fourier transforms are often used to calculate the frequency spectrum of a signal that changes over time. This kind of signal processing has many uses such as cryptography, oceanography, speech recognition, or handwriting recognition. Fourier transforms can also be used to solve differential equations.
Calculating a Fourier transform requires understanding of integration and imaginary numbers. Computers are usually used to calculate Fourier transforms of anything but the simplest signals. The Fast Fourier Transform is a method computers use to quickly calculate a Fourier transform.
52. Prime-counting functionâ€”
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by Ï€(x) (unrelated to the number Ï€).
Number of primes in up to the number x=x/lnx
53. MURPHYâ€™S LAW FORMULAâ€”
Here, PM is the Murphyâ€™s probability that something will go wrong. KM is Murphyâ€™s constant (equal to one) and FM is Murphyâ€™s factor, a very small number.
Murphy’s law is an adage or epigram that is typically stated as: “Anything that can go wrong will go wrong”.
54. SUMMATION FORMULAâ€”
In mathematics, summation (capital Greek sigma symbol: âˆ‘) is the addition of a sequence of numbers; the result is their sum or total. If numbers are added sequentially from left to right, any intermediate result is a partial sum, prefix sum, or running total of the summation.
The numbers to be summed (called addends, or sometimes summands) may be integers, rational numbers, real numbers, or complex numbers. Besides numbers, other types of values can be added as well: vectors, matrices, polynomials and, in general, elements of any additive group (or even monoid).
55. Logarithmic spiralâ€”A logarithmic spiral, equiangular spiral or growth spiral is a special kind of spiral spiral curve which often appears in nature. The logarithmic spiral was first described by descarte and later extensively investigated by Jakob bernoulli, who called it Spira mirabilis, “the marvelous spiral”.
Logarithmic spirals in nature
In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons:
â€¢The approach of a hawk to its prey. Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral’s pitch.
â€¢The approach of an insect to a light source. They are used to having the light source at a constant angle to their flight path. Usually the sun is the only light source and flying that way will result in a practically straight line.
â€¢The arms of spiral galaxies. Our own galaxy, the milky way. is believed to have four major spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees, an unusually small pitch angle for a galaxy such as the Milky Way. In general, arms in spiral galaxies have pitch angles ranging from about 10 to 40 degrees.
â€¢The arms of tropical cyclones, such as hurricanes.
â€¢Many biological structures including spider webs and the shells of mullosks. In these cases, the reason is the following: Start with any irregularly shaped two-dimensional figure F0. Expand F0 by a certain factor to get F1, and place F1 next to F0, so that two sides touch. Now expand F1 by the same factor to get F2, and place it next to F1 as before. Repeating this will produce an approximate logarithmic spiral whose pitch is determined by the expansion factor and the angle with which the figures were placed next to each other. This is shown for polygonal figures in the accompanying graphic.
56. Heron’s formulaâ€”
Heron’s formula states that the area of a triangle whose sides have lengths a, b, and c is
where s is the semiperimeter of the triangle; that is,
In geometry, Heron’s formula gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.
57. Quadratic equationâ€”
A quadratic equation is an equation in the form of ax2 + bx + c, where a is not equal to 0. It makes a parabola (a “U” shape) when graphed on a coordinate plane.
Where the letters are the corresponding numbers of the original equation, ax2 + bx + c = 0. Also, a cannot be 0 for the formula to work properly.
The factored form of this equation is y = a(x âˆ’ s)(x âˆ’ t), where s and t are the zeros, a is a constant, and y and the two xs are ordered pairs which satisfy the equation.
58. Euler lineâ€”
In geometry, the Euler line, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the 9-point circle of the triangle.
59. PYTHAGOREAN TRIPLES FORMULAâ€”
Euclid’s formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of integers m and n with m > n > 0. The formula states that the integers
form a Pythagorean triple
A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k.
60. Euler’s formulaâ€”
Eulerâ€™s formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential functions. Euler’s formula is ubiquitous in mathematics, physics, and engineering. The physicist richard feynmann called the equation “our jewel” and “the most remarkable formula in mathematics”.
Euler’s formula evaluates to
which is known as rulerâ€™s identity.
61. SIMPLEX METHODâ€”
In mathematical optimization, danzigâ€™s simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The simplex algorithm operates on linear programs in standard form:
with x=(x 1,â€¦,x n) the variables of the problem, c=(c 1,â€¦,c n)
are the coefficients of the objective function, A is a pÃ—n matrix, and b=
(b 1,â€¦,b p)
bjâ‰¥0. There is a straightforward process to convert any linear program into one in standard form so this results in no loss of generality.
In geometric terms, the feasible region defined by all values of
x such that Axâ‰¤b,x iâ‰¥0
is a (possibly unbounded) convex polytope. There is a simple characterization of the extreme points or vertices of this polytope, namely an element x=(x 1,â€¦,x n)
of the feasible region is an extreme point if and only if the subset of column vectors A i corresponding to the nonzero entries of x(x iâ‰ 0)
are linearly independent. In this context such a point is known as a basic feasible solution (BFS).
It can be shown that for a linear program in standard form, if the objective function has a maximum value on the feasible region then it has this value on (at least) one of the extreme points. This in itself reduces the problem to a finite computation since there is a finite number of extreme points, but the number of extreme points is unmanageably large for all but the smallest linear programs.
It can also be shown that if an extreme point is not a maximum point of the objective function then there is an edge containing the point so that the objective function is strictly increasing on the edge moving away from the point. If the edge is finite then the edge connects to another extreme point where the objective function has a greater value, otherwise the objective function is unbounded above on the edge and the linear program has no solution. The simplex algorithm applies this insight by walking along edges of the polytope to extreme points with greater and greater objective values. This continues until the maximum value is reached or an unbounded edge is visited, concluding that the problem has no solution. The algorithm always terminates because the number of vertices in the polytope is finite; moreover since we jump between vertices always in the same direction (that of the objective function), we hope that the number of vertices visited will be small.
The solution of a linear program is accomplished in two steps. In the first step, known as Phase I, a starting extreme point is found. Depending on the nature of the program this may be trivial, but in general it can be solved by applying the simplex algorithm to a modified version of the original program. The possible results of Phase I are either that a basic feasible solution is found or that the feasible region is empty. In the latter case the linear program is called infeasible. In the second step, Phase II, the simplex algorithm is applied using the basic feasible solution found in Phase I as a starting point. The possible results from Phase II are either an optimum basic feasible solution or an infinite edge on which the objective function is unbounded below.
62.Â PROOF INFINITE NUMBER OF PRIME NUMBERS
Euclid’s theorem is a fundamental statement in number theory that asserts that there are infinitely many primes numbers. There are several well-known proofs of the theorem.
Euclid offered a proof published in his work elements (Book IX, Proposition 20), which is paraphrased here.
Consider any finite list of prime numbers p1,Â p2,Â …,Â pn. It will be shown that at least one additional prime number not in this list exists. Let P be the product of all the prime numbers in the list: PÂ =Â p1p2…pn. Let qÂ =Â PÂ +Â 1. Then q is either prime or not:
â€¢If q is prime, then there is at least one more prime that is not in the list.
â€¢If q is not prime, then some prime factor p dividesÂ q. If this factor p were on our list, then it would divide P (since P is the product of every number on the list); but p divides PÂ +Â 1Â =Â q. If p divides P and q, then p would have to divide the difference of the two numbers, which is (PÂ +Â 1)Â âˆ’Â P or justÂ 1. Since no prime number divides 1, p cannot be on the list. This means that at least one more prime number exists beyond those in the list.
This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers.
Euclid is often erroneously reported to have proved this result by contradiction, beginning with the assumption that the finite set initially considered contains all prime numbers, or that it contains precisely the n smallest primes, rather than any arbitrary finite set of primes. Instead of a proof by contradiction, Euclid’s proof shows that no item on a finite list has a particular property. A contradiction is not inferred, but none of the items on the list can have the property of being a divisor of 1.
Several variations on Euclid’s proof exist, including the following:
The factorial n! of a positive integer n is divisible by every integer from 2 to n, as it is the product of all of them. Hence, n! + 1 is not divisible by any of the integers from 2 to n, inclusive (it gives a remainder of 1 when divided by each). Hence n! + 1 is either prime or divisible by a prime larger thanÂ n. In either case, for every positive integer n, there is at least one prime bigger thanÂ n. The conclusion is that the number of primes is infinite.
Paul erdos gave a third proof that also relies on the fundamental theorem of arithmetic. First note that every integer n can be uniquely written as
where r is square-free, or not divisible by any square numbers (let s2 be the largest square number that divides n and then let rÂ =Â n/s2). Now suppose that there are only finitely many prime numbers and call the number of prime numbersÂ k. As each of the prime numbers factorizes any squarefree number at most once, by the fundamental theorem of arithmetic, there are only 2k square-free numbers.
Now fix a positive integer N and consider the integers between 1 andÂ N. Each of these numbers can be written as rs2 where r is square-free and s2 is a square, like this:
( 1Ã—1, 2Ã—1, 3Ã—1, 1Ã—4, 5Ã—1, 6Ã—1, 7Ã—1, 2Ã—4, 1Ã—9, 10Ã—1, …)
There are N different numbers in the list. Each of them is made by multiplying a squarefree number, by a square number that is N or less. There are floor(âˆšN) such square numbers. Then, we form all the possible products of all squares less than N multiplied by all squarefrees everywhere. There are exactly 2kfloor(âˆšN) such numbers, all different, and they include all the numbers in our list and maybe more. Therefore, 2kfloor(âˆšN)Â â‰¥Â N.
Since this inequality does not hold for N sufficiently large, there must be infinitely many primes.
In the 1950s, Hillel furstenberg introduced a proof using Point-set topology.
63. Harmonic series (mathematics)â€”
In mathematics, the harmonic series is the divergent infinite series:
âˆ‘n=1 to âˆž1/n=1+1/2+1/3+1/4+1/5+â‹¯
Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are
1/2,1/3,1/4, etc., of the string’s fundamental wavelengths. Every term of the series after the first is the harmonic mean of the neighboring terms; the phrase harmonic mean likewise derives from music.
The harmonic series can be counterintuitive to students first encountering it, because it is a divergent series even though the limit of the nth term as n goes to infinity is zero. The divergence of the harmonic series is also the source of some apparent paradoxes. One example of these is the â€œworm on a rubberband”. Suppose that a worm crawls along an infinitely-elastic one-meter rubber band at the same time as the rubber band is uniformly stretched. If the worm travels 1 centimeter per minute and the band stretches 1 meter per minute, will the worm ever reach the end of the rubber band? The answer, counterintuitively, is “yes”, for after n minutes, the ratio of the distance travelled by the worm to the total length of the rubber band is
1/100âˆ‘k=1 to n1^k
(In fact the actual ratio is a little less than this sum as the band expands continuously.) The reason is that the band expands behind the worm also; eventually, the worm gets past the midway mark and the band behind expands increasingly more rapidly than the band in front.
Because the series gets arbitrarily large as n becomes larger, eventually this ratio must exceed 1, which implies that the worm reaches the end of the rubber band. However, the value of n at which this occurs must be extremely large: approximately e100, a number exceeding 1043 minutes (1037 years). Although the harmonic series does diverge, it does so very slowly.
Another problem involving the harmonic series is the jeep problem.
Another example is the block-stacking problem: given a collection of identical dominoes, it is clearly possible to stack them at the edge of a table so that they hang over the edge of the table without falling. The counterintuitive result is that one can stack them in such a way as to make the overhang arbitrarily large, provided there are enough dominoes.
A simpler example, on the other hand, is the swimmer that keeps adding more speed when touching the walls of the pool. The swimmer starts crossing a 10-meter pool at a speed of 2Â m/s, and with every cross, another 2Â m/s is added to the speed. In theory, the swimmer’s speed is unlimited, but the number of pool crosses needed to get to that speed becomes very large; for instance, to get to the speed of light (ignoring special relativity), the swimmer needs to cross the pool 150Â million times. Contrary to this large number, the time required to reach a given speed depends on the sum of the series at any given number of pool crosses (iterations):
10/2âˆ‘k=1 to n1^k.
Calculating the sum (iteratively) shows that to get to the speed of light the time required is only 94Â seconds. By continuing beyond this point (exceeding the speed of light, again ignoring special relativity), the time taken to cross the pool will in fact approach zero as the number of iterations becomes very large, and although the time required to cross the pool appears to tend to zero (at an infinite number of iterations), the sum of iterations (time taken for total pool crosses) will still diverge at a very slow rate.
64. EULER SUMSâ€”
e^x=âˆ‘n=0 to âˆž to x^n/n!=limnâ†’âˆž(1/0!+x1!+x^2/2!+â‹¯+x^n/n!)
âˆ‘n=1 to âˆž1/n^2=lim nâ†’âˆž(1/1^2+1/2^2+1/3^2+â‹¯+1/n^2)=Ï€^2/6
65. FORMULA FOR SOLUTION OF CUBIC EQUATIONâ€”
In algebra, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d
in which a is nonzero.
Setting f(x) = 0 produces a cubic equation of the form
The solutions of this equation are called roots of the polynomial f(x). If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root (this is true for all odd degree polynomials). All of the roots of the cubic equation can be found algebraically. (This is also true of a quadratic or quartic (fourth degree) equation, but no higher-degree equation, by the abel-ruffini theorem). The roots can also be found trigonometrically. Alternatively, numeric approximates of the roots can be found using root-finding theorem like newtonâ€™s method.
The coefficients do not need to be complex numbers. Much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the same field as the coefficients. For example, some cubic equations with rational coefficients have roots that are non-rational (and even non-real) complex numbers.
66. SOLUTION TO QUARTIC EQUATIONâ€”
In algebra, a quartic function is a function of the form
where a is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial.
Sometimes the term biquadratic is used instead of quartic, but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form
A quartic equation, or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form
where a â‰ 0.
The derivative of a quartic function is a cubic function.
Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If a is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if a is negative, it decreases to negative infinity and has a global maximum. In both cases it may or may not have another local maximum and another local minimum.
The degree four (quartic case) is the highest degree such that every polynomial equation can be solved by radicals.
67. Quintic functionâ€”
In algebra, a quintic function is a function of the form
where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other words, a quintic function is defined by a polynomials of degree five.
If a is zero but one of the coefficients b, c, d, or e is non-zero, the function is classified as either a quartic function, cubic function, quadratic function or linear function.
Because they have an odd degree, normal quintic functions appear similar to normal cubic function when graphed, except they may possess an additional local maximun and local minimum each. The derivative of a quintic function is a quartic function.
Setting g(x) = 0 and assuming a â‰ 0 produces a quintic equation of the form:
Solving quintic equations in terms of radicals was a major problem in algebra, from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved, with the abel-ruffini theorem.
Finding the roots of a given polynomial has been a prominent mathematical problem. Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression for general quintic equations over the rationals in terms of radicals. This also holds for equations of higher degrees. Some quintics may be solved in terms of radicals. However, the solution is generally too complex to be used in practice. Instead, numerical approximations are calculated using Root-finding algorithms for polynomials. Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible, such as x5 âˆ’ x4 âˆ’ x + 1 = (x2 + 1)(x + 1)(x âˆ’ 1)2. For characterizing solvable quintics, and more generally solvable polynomials of higher degree, evariste galois developed techniques which gave rise to group theory and galois theory. Applying these techniques, arthur caylay found a general criterion for determining whether any given quintic is solvable.
68. Lorentz forceâ€”
In physics (particularly in electromagnetism) the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge q moving with velocity v in the presence of an electric field E and a magnetic field B experiences a force
(in SI units).
69. Euler-lagrange formula-
In the calculus of variations, the Eulerâ€“Lagrange equation, Euler’s equation, or Lagrange’s equation (although the latter name is ambiguousâ€”see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary. It was developed by Swiss mathematician Leonhard Euler and Italian-French mathematician Joseph-Louis Lagrange in the 1750s.
70. Euler product formulaâ€”In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Euler. This series and its continuation to the entire complex plane would later become known as the Riemann zeta function.
71. Euler-maclaurin formulaâ€”In mathematics, the Eulerâ€“Maclaurin formula provides a powerful connection between integrals (see calculus) and sums. It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus. For example, many asymptotic expansions are derived from the formula, and Faulhaber’s formula for the sum of powers is an immediate consequence.
(pi is equal to the circumference divided by the diameter).
Pi is an endless string of numbers
Pi (or Ï€) is a mathematical constant. It is the ratio of the distance around a circle to the circle’s diameter. This produces a number, and that number is always the same. However, the number is rather strange. The number starts 3.141592……. and continues without end. Numbers like this are called irrational numbers.
The diameter is the longest straight line which can be fitted inside a circle. It passes through the center of the circle. The distance around a circle is known as the circumference. Even though the diameter and circumference are different for different circles, the number pi remains constant: its value never changes. This is because the relationship between the circumference and diameter is always the same.
A mathematician named Lambert also showed in 1761 that the number pi was irrational; that is, it cannot be written as a fraction by normal standards. Another mathematician named Lindeman was also able to show in 1882 that pi was part of the group of numbers known as transcendental, which are numbers that cannot be the solution to a polynomial equation.
Pi can also be used for figuring out many other things beside circles. The properties of pi have allowed it to be used in many other areas of math besides geometry, which studies shapes. Some of these areas are complex analysis, trigonometry, and series.
Today, there are different ways to calculate many digits of Ï€. This is of limited use though.Pi can sometimes be used to work out the area or the circumference of any circle. To find the circumference of a circle, use the formula C (circumference) = Ï€ times diameter. To find the area of a circle, use the formula Ï€ (radiusÂ²). This formula is sometimes written as
A=Ï€r^2, where r is the variable for the radius of any circle and A is the variable for the area of that circle.
To calculate the circumference of a circle with an error of 1Â mm:
â€¢4 digits are needed for a radius of 30 meters
â€¢10 digits for a radius equal to that of the earth
â€¢15 digits for a radius equal to the distance from the earth to the sun.
People generally celebrate March 14 as pi day because March 14 is also written as 3/14, which represents the first three numbers 3.14 in the approximation of pi.
73. Exponential functionâ€”
In mathematics, an exponential function is a function that quickly grows. More precisely, it is the function
exp(x)=e^x, where e is rulerâ€™s constant, an irrational number that is approximately 2.71828.
74. Natural logarithmâ€”
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. The natural logarithm of x is the power to which e would have to be raised to equal x.
75. Conic sectionsâ€”
(g=x coordinate, h=y coordinate)
(a=x, b=y coordinates, or a=y, b=x coordinates)
(a=x, b=y coordinates, or a=y, b=x coordinates)
76. Exponential growth and decayâ€”
A=starting number of for example bacteria, t=length of growth time, k=constant, y=number of bacteria after t time
77. Calculating an orbit I.e. of a cometâ€”
Calculations: orbit, period of orbit, perihelion, aphelion and eccentricityâ€”
(for example a comet)â€”
Use ellipse formulaâ€”x^2/a^2+y^2/b^2=1
Then calculate from 2 coordinates in AUs with formulaâ€”
x^2 x b^2 + y^2 x a^2=a^2 x b^2
find a and bÂ (the closest and furthest approaches)
Period years of orbit^3=distance (a from above)^2
Period=cuberoot(distance AUs of â€˜aâ€™ above)^2
c=(a^2-b^2)^1/2Â (c=distance from focus to center of ellipse)
perihelion=A x (1-eccentricity)
aphelion=A x (1+ eccenticity)
A (semimajor axis)=(perihelion + aphelion)/2
To find formula for the orbit, use ellipse formulaâ€”
x^2/a^2+y^2/b^2=1, then use formulaâ€”
x^2 x b^2+y^2 x a^2=a^2 x b^2,
Use 2 location coordinates from the orbit, plug in one of the coordinates
Into the 2nd formula, then plug in the 2nd coordinates into the same
formula. Subtract one of the resulting formulas from the other resulting
formula, then solve for a or bÂ with the formula that results from the
subtraction. Plug in the solution to a or b that was solved into one of the
Pre-subtraction formulas to find the a or b that has not been found yet.
Now, we have the a and b constants, so we plug them into the ellipse
Formula, and thus have the equation for the orbit of the stellar body,
I.e. a comet.
78. Interesting math example #1
79. Interesting math example #2
80. Interesting math example #3
In mathematics, 0.999… (also written 0.9, among other ways), denotes the repeating decimal consisting of infinitely many 9 after the decimal point (and one 0 before it). This repeating decimal represents the smallest number no less than all decimal number 0.9, 0.99, 0.999, etc. This number can be shown to equal 1. In other words, “0.999…” and “1” represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proof. The technique used depends on target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999… is commonly defined. (In other systems, 0.999… can have the same meaning, a different definition, or be undefined.) More generally, every nonzero terminating decimals has two equal representations (for example, 8.32 and 8.31999…), a property true of all base representations. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasonsâ€”such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplinesâ€”math students can find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education.