John hebert. 3/22/18
1. Dirac’s equation—The Dirac equation is the relativistic description of an electron. The non-relativistic description of an electron is described by the Pauli-Schroedinger equation.
2. Einstein’s field equation—
Einstein field equations. The Einstein field equations (EFE; also known as Einstein’s equations) comprise the set of 10 equations in Albert Einstein’s general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
3. Maxwell’s equations—
Maxwell’s equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. … The microscopic Maxwell equations have universal applicability, but are unwieldy for common calculations.
4. General relativity—Einstein field equations. The Einstein field equations (EFE; also known as Einstein’s equations) comprise the set of 10 equations in Albert Einstein’s general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.
5. Special relativity—To derive the equations of special relativity, one must start with two postulates: The laws of physics are invariant under transformations between inertial frames. … The speed of light in a vacuum is measured to be the same by all observers in inertial frames.
E = mc2
dilation, length contraction, and more.
6. Schrodinger’s equation—
The Schrodinger equation is used to find the allowed energy levels of quantum mechanical systems (such as atoms, or transistors). The associated wavefunction gives the probability of finding the particle at a certain position. … The solution to this equation is a wave that describes the quantum aspects of a system.
7. Uncertainty principle—Uncertainty principle, also called Heisenberg uncertainty principle or indeterminacy principle, statement, articulated (1927) by the German physicist Werner Heisenberg, that the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. the position and the velocity of an object cannot both be measured exactly, at the same time, even in theory. The very concepts of exact position and exact velocity together, in fact, have no meaning in nature.
8. Gibb’s statistical mechanics—Statistical mechanics is a branch of theoretical physics that uses probability theory to study the average behaviour of a mechanical system whose exact state is uncertain. In mathematical physics, especially as introduced into statistical mechanics and thermodynamics by J. Willard Gibbs in 1902, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system. A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics.
9. Stephan-Boltzmann law—
Stefan-Boltzmann law, statement that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature.
E = mc^2 definition. An equation derived by the twentieth-century physicist Albert Einstein, in which E represents units of energy, m represents units of mass, and c^2 is the speed of light squared, or multiplied by itself.
11. Laplace equation—Laplace’s equation and Poisson’s equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace’s equation is known as potential theory. 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.
In mathematics, Laplace’s equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:
∇2φ=0 or Δφ=0
where ∆ = ∇2 is the Laplace operator (see below) and
φ is a scalar function.
12. De broglie relation-matter wave—Matter waves are referred to as de Broglie waves. The de Broglie wavelength is the wavelength, λ, associated with a massive particle and is related to its momentum, p, through the Planck constant, h: … The wave-like behavior of matter is crucial to the modern theory of atomic structure and particle physics. the relations λ = h/p and f = E/h are called the de Broglie
13. Navier-stokes equations—Navier-Stokes Equations. On this slide we show the three-dimensional unsteady form of the Navier-Stokes Equations. These equations describe how the velocity, pressure, temperature, and density of a moving fluid are related. … This area of study is called Computational Fluid Dynamics or CFD. describe the motion of viscous fluid substances.
14. Riemann zeta function—The Riemann zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day. The Riemann zeta function is defined over the complex plane for one complex variable.
15. Noether theorem—Noether’s theorem is an amazing result which lets physicists get conserved quantities from symmetries of the laws of nature. Time translation symmetry gives conservation of energy; space translation symmetry gives conservation of momentum; rotation symmetry gives conservation of angular momentum, and so on.
The momentum of our particle is defined to be
p = dL/dq’
The force on it is defined to be
F = dL/dq
The equations of motion – the so-called Euler-Lagrange equations – say that the rate of change of momentum equals the force:
p’ = F
That’s how Lagrangians work!
16. Euler-lagrange equation—In broad strokes, the Euler-Lagrange equations are used in physics to find stationary points of the action S. The action is defined as a functional of the Lagrangian. A functional is a sort of function of a function. The Euler-Lagrange differential equation is the fundamental equation of calculus of variations.
17. Hamilton quanternion—In mathematics, quaternions are a non-commutative number system that extends the complex numbers.
18. Standard model—The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.
19. Lagrange formula—
In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points,yj) with no two xj values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value xj the corresponding value yj (i.e. the functions coincide at each point). The interpolating polynomial of the least degree is unique, however, and since it can be arrived at through multiple methods, referring to “the Lagrange polynomial” is perhaps not as correct as referring to “the Lagrange form” of that unique polynomial. Since Lagrange’s interpolation is also an Nth degree polynomial approximation to f(x) and the Nth degree polynomial passing through (N+1) points is unique hence the Lagrange’s and Newton’s divided difference approximations are one and the same. In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of points with no two values equal, the Lagrange polynomial is the polynomial of lowest degree that assumes at each value the corresponding value (i.e. the functions coincide at each point).
20. cantor inequality—In elementary set theory, Cantor’s theorem is a fundamental result that states that, for any set A, the set of all set of A (the power set of A, denoted by P(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 2^n subsets,so that if card(A)=n, then card(P(A))=2n(A))=2^n, and the theorem holds because
2^n>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 ℵ0 = card(ℕ), is uncountably infinite and has the same size as the set of real numbers, a cardinality often referred to as the cardinality of the continuum: 𝔠 = card(ℝ) = card(𝒫(ℕ)). The relationship between these cardinal numbers is often expressed symbolically by the equality c=2^ℵ0.
21. Riemann hypothesis—In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Riemann zeta function, function useful in number theory for investigating properties of prime numbers. Written as ζ(x), it was originally defined as the infinite series ζ(x) = 1 + 2−x + 3−x + 4−x + ⋯. When x = 1, this series is called the harmonic series, which increases without bound—i.e., its sum is infinite. the Riemann hypothesis is a deep mathematical conjecture which states that the nontrivial Riemann zeta function zeros, i.e., the values of other than , , , … such that (where is the Riemann zeta function) all lie on the “critical line” (where denotes the real part of ).
22. Hawking-Bekenstein entropy formula—Starting from theorems proved by Stephen Hawking, Jacob Bekenstein conjectured that the black hole entropy was proportional to the area of its event horizon divided by the Planck area. … This is often referred to as the Bekenstein–Hawking formula.
23. Heat equation—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.
24. wave equation—The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as sound waves, light waves and water waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.The wave equation is the important partial differential equation that describes propagation of waves with speed . The form above gives the wave equation in three-dimensional space. As with all partial differential equations, suitable initial and/or boundary conditions must be given to obtain solutions to the equation for particular geometries and starting conditions.
25. poisson equation— In mathematics, Poisson’s equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. … It is a generalization of Laplace’s equation, which is also frequently seen in physics.
26. Wave-particle duality—Wave–particle duality is the concept in quantum mechanics that every particle or quantic entity may be partly described in terms not only of particles, but also of waves. It expresses the inability of the classical concepts “particle” or “wave” to fully describe the behavior of quantum-scale objects.
27. fundamental theorem of calculus—The first part of the theorem, sometimes called the first fundamental theorem of calculus, states that one of the antiderivatives (also called indefinite integral), say F, of some function f may be obtained as the integral of f with a variable bound of integration. This result, while taught early in elementary calculus courses, is actually a very deep result connecting the purely algebraic indefinite integral and the purely analytic (or geometric) definite integral.
28. Pythagorean theorem—In mathematics, the Pythagorean theorem, also known as Pythagoras’ theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
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 to their topology. The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian curvature of an embedded triangle in terms of the total geodesic curvature of the boundary and the jump angles at the corners.
30. universal law of gravitation—Newton’s law of universal gravitation states that a particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
31. Newton’s 2nd law of motion—Newton’s second law of motion can be formally stated as follows: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
Fnet = m • a
1 Newton = 1 kg • m/s2
32. kinetic energy—energy that a body possesses by virtue of being in motion.
1 Joule = 1 kg • m2/s2
33. Potential energy—the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors.
PEgrav = mass • g • height
PEgrav = m *• g • h
In the above equation, m represents the mass of the object, h represents the height of the object and g represents the gravitational field strength (9.8 N/kg on Earth) – sometimes referred to as the acceleration of gravity.
34. 2nd law of thermodynamics—The Second Law of Thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative. Second Law of Thermodynamics – Increased Entropy
The Second Law of Thermodynamics is commonly known as the Law of Increased Entropy. While quantity remains the same (First Law), the quality of matter/energy deteriorates gradually over time. How so? Usable energy is inevitably used for productivity, growth and repair. In the process, usable energy is converted into unusable energy. Thus, usable energy is irretrievably lost in the form of unusable energy. “Entropy” is defined as a measure of unusable energy within a closed or isolated system (the universe for example). As usable energy decreases and unusable energy increases, “entropy” increases. Entropy is also a gauge of randomness or chaos within a closed system. As usable energy is irretrievably lost, disorganization, randomness and chaos increase.
35. principle of least action—The principle of least action – or, more accurately, the principle of stationary action – 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.
36. Spherical harmonics—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. The spherical harmonics are the angular portion of the solution to Laplace’s equation in spherical coordinates where azimuthal symmetry is not present. Some care must be taken in identifying the notational convention being used. In this entry, is taken as the polar (colatitudinal) coordinate with , and as the azimuthal (longitudinal) coordinate with . This is the convention normally used in physics, as described by Arfken (1985) and the Wolfram Language (in mathematical literature, usually denotes the longitudinal coordinate and the colatitudinal coordinate). Spherical harmonics are implemented in the Wolfram Language as SphericalHarmonicY[l, m, theta, phi]. Spherical harmonics satisfy the spherical harmonic differential equation, which is given by the angular part of Laplace’s equation in spherical coordinates.
37. Cauchy residue theorem—
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.
38. Callen-Symanzik equation—In physics, the Callan–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.
the Callan-Symanzik equation can be put in the conventional form:
β(g) being the beta function.
In quantum electrodynamics this equation takes the form [M∂∂M+β(e)∂/∂e+nγ2+mγ3]G(n,m)(x1,x2,…,xn;y1,y2,…,ym;M,e)=0
where n and m are the numbers of electron and photon fields, respectively, for which the correlation function G(n,m) is defined.
39. Minimal surface equation—In mathematics, a minimal surface is a surface that locally minimizes its area. … The term “minimal surface” is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Minimal surfaces are defined as surfaces with zero mean curvature. Finding a minimal surface of a boundary with specified constraints is a problem in the calculus of variations and is sometimes known as Plateau’s problem. Minimal surfaces may also be characterized as surfaces of minimal surface area for given boundary conditions. A plane is a trivial minimal surface, and the first nontrivial examples (the catenoid and helicoid) were found by Meusnier in 1776 (Meusnier 1785). The problem of finding the minimum bounding surface of a skew quadrilateral was solved by Schwarz in 1890 (Schwarz 1972).
Note that while a sphere is a “minimal surface” in the sense that it minimizes the surface area-to-volume ratio, it does not qualify as a minimal surface in the sense used by mathematicians. Euler proved that a minimal surface is planar iff its Gaussian curvature is zero at every point so that it is locally saddle-shaped. The existence of a solution to the general case was independently proven by Douglas (1931) and Radó (1933), although their analysis could not exclude the possibility of singularities. Osserman (1970) and Gulliver (1973) showed that a minimizing solution cannot have singularities. The only known complete (boundaryless), embedded (no self-intersections) minimal surfaces of finite topology known for 200 years were the catenoid, helicoid, and plane. Hoffman discovered a three-ended genus 1 minimal embedded surface, and demonstrated the existence of an infinite number of such surfaces. A four-ended embedded minimal surface has also been found. L. Bers proved that any finite isolated singularity of a single-valued parameterized minimal surface is removable.A surface can be parameterized using a isothermal parameterization. Such a parameterization is minimal if the coordinate functions are harmonic, i.e., are analytic.
40. Euler 9 point center—
The Euler lines of the four triangles formed by an orthocentric system (a set of four points such that each is the orthocenter of the triangle with vertices at the other three points) are concurrent at the nine-point center common to all of the triangles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: The midpoint of each side of the triangle.
41. Mandelbrot set—a particular set of complex numbers that has a highly convoluted fractal boundary when plotted. As a consequence of the definition of the Mandelbrot set, there is a close correspondence between the geometry of the Mandelbrot set at a given point and the structure of the corresponding Julia set. For instance, a point is in the Mandelbrot set exactly when the corresponding Julia set is connected. The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable.
42. Yang-Baxter equation—In one dimensional quantum systems, is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–Baxter equation also shows up when discussing knot theory and the braid groups where corresponds to swapping two strands. It explains the mathematical theory of knots, among other things. In physics, the Yang–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 (R⊗1)(1⊗R)(R⊗1)=(1⊗R)(R⊗1)(1⊗R) In one dimensional quantum systems,R is the scattering matrix and if it satisfies the Yang–Baxter equation then the system is integrable. The Yang–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 Yang–Baxter equation enforces that both paths are the same.
43. Divergence theorem—
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.
44. Baye’s theorem—
Bayes’ theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. It follows simply from the axioms of conditional probability, but can be used to powerfully reason about a wide range of problems involving belief updates. Bayesian inference is a method of statistical inference in which Bayes’ theorem is used to update the probability for a hypothesis as more evidence or information becomes available. Bayesian inference is an important technique in statistics, and especially in mathematical statistics.
45. logistic map—
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations.
46. Einstein’s law of velocity addition—
Einstein Velocity Addition. The relative velocity of any two objects never exceeds the velocity of light. Applying the Lorentz transformation to the velocities, expressions are obtained for the relative velocities as seen by the different observers. They are called the Einstein velocity addition relationships.
47. Photoelectric effect formula—
The photoelectric equation involves; h = the Plank 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).
48. Faraday law—
Faraday’s law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. Faraday’s law of induction is a basic law of electromagnetism predicting how a magnetic field will interact with an electric circuit to produce an electromotive force (EMF)—a phenomenon called electromagnetic induction. It is the fundamental operating principle of transformers, inductors, and many types of electrical motors, generators and solenoids.
Faraday’s Law, which is given in Equation :
49. Cauchy momentum equation—
The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum. By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. … The stress terms p and τ are yet unknown, so the general form of the equations of motion is not usable to solve problems.
50. De moivre’s theorem—
The process of mathematical induction can be used to prove a very important theorem in mathematics known as De Moivre’s theorem. If the complex number z = r(cos α + i sin α), then. The preceding pattern can be extended, using mathematical induction, to De Moivre’s theorem.
51. Fourier transform—The Fourier transform is a mathematical technique that allows an MR signal to be decomposed into a sum of sine waves of different frequencies, phases, and amplitudes.
The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of time. … After performing the desired operations, transformation of the result can be made back to the time domain.
52. prime counting function—
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 π(x) (unrelated to the number π). The prime counting function is the function giving the number of primes less than or equal to a given number (Shanks 1993, p. 15). For example, there are no primes , so . There is a single prime (2) , so . There are two primes (2 and 3) , so . And so on. The notation for the prime counting function is slightly unfortunate because it has nothing whatsoever to do with the constant . This notation was introduced by number theorist Edmund Landau in 1909 and has now become standard. In the words of Derbyshire (2004, p. 38), “I am sorry about this; it is not my fault. You’ll just have to put up with it.”
53. Murphy’s law—The mathematical statement of Murphy’s Law, as used in scientific communities, is tremendously complex. But the common form, “everything that can go wrong will”, is fairly accurate and more than sufficient for most applications. The short answer is: yes, Murphy’s Law is real. “If anything can go wrong, it will go wrong. “ is one way to express the famous adage known by such names as Murphy’s Law, Finagle’s Law, and Sod’s Law. Some people consider it a myth while others take it seriously. British mathematician Philip Obadya. working with colleagues David Lewis and Keylan Leyser, came up with a formula that statistically calculates the likelihood of this law. 54. Summation formula—Summation Notation. Often mathematical formulae require the addition of many variables Summation or sigma notation is a convenient and simple form of shorthand used to give a concise expression for a sum of the values of a variable. Let x1, x2, x3, …xn denote a set of n numbers.
54. Logarithmic spiral—
A logarithmic spiral, equiangular spiral or growth spiral is a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, “the marvelous spiral”. This spiral is related to Fibonacci numbers, the golden ratio, and the golden rectangle, and is sometimes called the golden spiral. The logarithmic spiral can be constructed from equally spaced rays by starting at a point along one ray, and drawing the perpendicular to a neighboring ray. As the number of rays approaches infinity, the sequence of segments approaches the smooth logarithmic spiral (Hilton et al. 1997, pp. 2-3). The logarithmic spiral was first studied by Descartes in 1638 and Jakob Bernoulli. Bernoulli was so fascinated by the spiral that he had one engraved on his tombstone (although the engraver did not draw it true to form) together with the words “eadem mutata resurgo” (“I shall arise the same though changed”). Torricelli worked on it independently and found the length of the curve (MacTutor Archive).
55. Heron’s formula—
Heron’s formula is named after Hero of Alexendria, a Greek Engineer and Mathematician in 10 – 70 AD. You can use this formula to find the area of a triangle using the 3 side lengths.
57. Quadratic equation—
While factoring may not always be successful, the Quadratic Formula can always find the solution. The Quadratic Formula uses the “a”, “b”, and “c” from “ax2 + bx + c”, where “a”, “b”, and “c” are just numbers; they are the “numerical coefficients” of the quadratic equation they’ve given you to solve.
58. Euler line
59. Pythagorean triple formula—
The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a^2 + b^2 = c^2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples.
60. Euler’s formula—
Twenty Proofs of Euler’s Formula: V-E+F=2. … This page lists proofs of the Euler formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges. Symbolically V−E+F=2. For instance, a tetrahedron has four vertices, four faces, and six edges; 4-6+4=2.
61. euler’s identity—
π is pi, the ratio of the circumference of a circle to its diameter. Euler’s identity is named after the Swiss mathematician Leonhard Euler.
62. Simplex method—In mathematical optimization, Dantzig’s simplex algorithm (or simplex method) is a popular algorithm for linear programming.
The Simplex method is a search procedure that sifts through the set of basic feasible solutions, one at a time, until the optimal basic feasible solution (whenever it exists) is identified. The method is essentially an efficient implementation of both Procedure Search and Procedure Corner Points discussed in the previous section.
63. Proof of infinity of prime numbers
64. Harmonic series—
In mathematics, the harmonic series is the divergent infinite series: Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 12, 13, 14, etc., of the string’s fundamental wavelength. In mathematics, the harmonic series is the divergent infinite series:
65. Euler sums—Given a series ∑an, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.
66. newton’s method—
In numerical analysis, Newton’s method (also known as the Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function. It is one example of a root-finding algorithm. Sometimes we are presented with a problem which cannot be solved by simple algebraic means. For instance, if we needed to find the roots of the polynomial , we would find that the tried and true techniques just wouldn’t work. However, we will see that calculus gives us a way of finding approximate solutions.
67. Newton’s law of cooling—
Newton’s Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature (i.e. the temperature of its surroundings). Newton’s Law of Cooling Formula. Sir Isaac Newton created a formula to calculate the temperature of an object as it loses heat. The heat moves from the object to its surroundings. The rate of the temperature change is proportional to the temperature difference between the object and its surroundings.
68. Laplace transform—In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace (/ləˈplɑːs/). It takes a function of a real variable t (often time) to a function of a complex variable s (frequency).The Laplace transform is very similar to the Fourier transform. While the Fourier transform of a function is a complex function of a real variable (frequency), the Laplace transform of a function is a complex function of a complex variable. Laplace transforms are usually restricted to functions of t with t ≥ 0. A consequence of this restriction is that the Laplace transform of a function is a holomorphic function of the variable s. Unlike the Fourier transform, the Laplace transform of a distribution is generally a well-behaved function. Also techniques of complex variables can be used directly to study Laplace transforms. As a holomorphic function, the Laplace transform has a power series representation. This power series expresses a function as a linear superposition of moments of the function. This perspective has applications in probability theory.
The Laplace transform is invertible on a large class of functions. The inverse Laplace transform takes a function of a complex variable s (often frequency) and yields a function of a real variable t (time). Given a simple mathematical or functional description of an input or output to a system, the Laplace transform provides an alternative functional description that often simplifies the process of analyzing the behavior of the system, or in synthesizing a new system based on a set of specifications. So, for example, Laplace transformation from the time domain to the frequency domain transforms differential equations into algebraic equations and convolution into multiplication. It has many applications in the sciences and technology.The Laplace transform is an integral transform perhaps second only to the Fourier transform in its utility in solving physical problems. The Laplace transform is particularly useful in solving linear ordinary differential equations such as those arising in the analysis of electronic circuits.
69. Euler product formula—
For Euler, s is an integer. It turns out that when s=1, ζ(s) = the harmonic series and is therefore divergent. When s ≠ 1, it turns out the the zeta function is convergent, that is, it has a finite limit. In this way, Euler came up with his Product Formula.
if a is a multiplicative function, then
summation n to infinity=a(n)n^-s
70. Euler-maclaurin formulas—
In mathematics, the Euler–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.
71. boyle’s law—
Gas laws, Laws that relate the pressure, volume, and temperature of a gas. Boyle’s law—named for Robert Boyle—states that, at constant temperature, the pressure P of a gas varies inversely with its volume V, or PV = k, where k is a constant.
72. Charlie’s law—
Charles’s law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles’s law is: When the pressure on a sample of a dry gas is held constant, the Kelvin temperature and the volume will be directly related.
Charles’ Law is a special case of the ideal gas law. It states that the volume of a fixed mass of a gas is directly proportional to the temperature. This law applies to ideal gases held at a constant pressure, where only the volume and temperature are allowed to change.
Charles’ Law is expressed as:
Vi/Ti = Vf/Tf
Vi = initial volume
Ti = initial absolute temperature
Vf = final volume
Tf = final absolute temperature
It is extremely important to remember the temperatures are absolute temperatures measured in Kelvin, NOT °C or °F.
73. Gay Lussac law—
Gay-Lussac’s law, Amontons’ law or the pressure law was found by Joseph Louis Gay-Lussac in 1809. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is directly proportional to its absolute temperature.
The mathematical form of Gay-Lussac’s Law is:
pi and pf are initial and final pressures
ti and tf are initial and final temperatures
This means that the pressure-temperature fraction will always be the same value if the volume and amount remain constant.
74. Ideal gas law—
The volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in Kelvin. The relationship for these variables, P V = n R T, where R is known as the gas constant, is called the ideal gas law or equation of state.
75. Lagrange multipliers—
In mathematical optimization, the method of Lagrange multipliers (named after Joseph-Louis Lagrange) is a strategy for finding the local maxima and minima of a function subject to equality constraints.
76. laplacian—In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇ · ∇, ∇2, or Δ.
The Laplacian is extremely important in mechanics, electromagnetics, wave theory, and quantum mechanics, and appears in Laplace’s equation, the Helmholtz differential equation, the wave equation, and the Schrödinger equation
77. Exponential growth and decay formula—
y=A x e^(p x t)
78. Infinite series formula—An infinite series that is geometric. An infinite geometric series converges if its common ratio r satisfies –1 < r < 1. infinite Geometric Series only. A series can converge or diverge. A series that converges has a finite limit, that is a number that is approached. A series that diverges means either the partial sums have no limit or approach infinity. … < 1, then the series will converge.
• SUM OF AN INFINITE GEOMETRIC SERIES If |r| < 1, the infinite Geometric Series: t1 + t1r + t1r2 + . . . + t1rn + . . . converges to the sum s = t1 / (1 – r). If |r| > 1, and t1 does not = 0, then the series diverges.
79. Infinite arithmetic series formula—Arithmetic Progression. A sequence such as 1, 5, 9, 13, 17 or 12, 7, 2, –3, –8, –13, –18 which has a constant difference between terms. The first term is a1, the common difference is d, and the number of terms is n. Explicit Formula: an = a1 + (n – 1)d.
80. Euler-lagrange formula—
In the calculus of variations, the Euler–Lagrange equation, Euler’s equation, or Lagrange’s equation (although the latter name is ambiguous—see disambiguation page), is a second-order partial differential equation whose solutions are the functions for which a given functional is stationary.
81. Lorentz equation—
Lorentz force, the force exerted on a charged particle q moving with velocity v through an electric E and magnetic field B. The entire electromagnetic force F on the charged particle is called the Lorentz force (after the Dutch physicist Hendrik A. Lorentz) and is given by F = qE + qv × B.
82. Euler product formula—
For Euler, s is an integer. It turns out that when s=1, ζ(s) = the harmonic series and is therefore divergent. When s ≠ 1, it turns out the the zeta function is convergent, that is, it has a finite limit. In this way, Euler came up with his Product Formula.
83. Conic sections–
84. ‘e’—The number e is a mathematical constant, approximately equal to 2.71828, which appears in many different settings throughout mathematics. It was discovered by the Swiss mathematician Jacob Bernoulli while studying compound interest, where e arises as the limit of (1 + 1/n)n as n approaches infinity. The number e can also be calculated as the sum of the infinite series e=∑n=0∞1/n!=1/1+1/1+1/1⋅2+1/1⋅2⋅3+⋯
Also called Euler’s number after the Swiss mathematician Leonhard Euler, e is different from γ, the Euler–Mascheroni constant, which is sometimes called simply Euler’s constant. Occasionally, the number e is termed Napier’s constant, but Euler’s choice of the symbol e is said to have been retained in his honor.[better source needed]
The number e is of eminent importance in mathematics, alongside 0, 1, π and i. All five of these numbers play important and recurring roles across mathematics, and are the five constants appearing in one formulation of Euler’s identity. Like the constant π, e is irrational: it is not a ratio of integers. Also like π, e is transcendental: it is not a root of any non-zero polynomial with rational coefficients.
85. .euler’s identity—
π is pi, the ratio of the circumference of a circle to its diameter. Euler’s identity is named after the Swiss mathematician Leonhard Euler.
86. Natural logarithm—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 to2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x.
87. Cubic equation—
Every cubic equation (1), ax3 + bx2 + cx +d = 0, with real coefficients and a ≠ 0, has three solutions (some of which may equal each other if they are real, and two of which may be complex non-real numbers) and at least one real solution r1, this last assertion being a consequence of the intermediate value theorem.
The cubic formula is the closed-form solution for a cubic equation, i.e., the roots of a cubic polynomial.
The solution to the cubic (as well as the quartic) was published by Gerolamo Cardano (1501-1576) in his treatise Ars Magna. However, Cardano was not the original discoverer of either of these results. The hint for the cubic had been provided by Niccolò Tartaglia, while the quartic had been solved by Ludovico Ferrari. However, Tartaglia himself had probably caught wind of the solution from another source. The solution was apparently first arrived at by a little-remembered professor of mathematics at the University of Bologna by the name of Scipione del Ferro (ca. 1465-1526). While del Ferro did not publish his solution, he disclosed it to his student Antonio Maria Fior (Boyer and Merzbach 1991, p. 283). This is apparently where Tartaglia learned of the solution around 1541.
88. Quartic equation solution—A quartic equation is a fourth-order polynomial equation. the term for a quartic equation having no cubic term, i.e., a quadratic equation in . Ferrari was the first to develop an algebraic technique for solving the general quartic, which was stolen and published in Cardano’s Ars Magna in 1545 (Boyer and Merzbach 1991, p. 283).
89. Quintic equation solution—Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Abel (Abel’s impossibility theorem) and Galois. However, certain classes of quintic equations can be solved in this manner.
90. Pi—The number π (/paɪ/) is a mathematical constant. Originally defined as the ratio of a circle’s circumference to its diameter, it now has various equivalent definitions and appears in many formulas in all areas of mathematics and physics. It is approximately equal to 3.14159. It has been represented by the Greek letter “π” since the mid-18th century, though it is also sometimes spelled out as “pi”.
Being an irrational number, π cannot be expressed exactly as a common fraction (equivalently, its decimal representation never ends and never settles into a permanent repeating pattern). Still, fractions such as 22/7 and other rational numbers are commonly used to approximate π. The digits appear to be randomly distributed. In particular, the digit sequence of π is conjectured to satisfy a specific kind of statistical randomness, but to date, no proof of this has been discovered. Also, π is a transcendental number; that is, a number that is not the root of any non-zero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straightedge. Ancient civilizations required fairly accurate computed values for π for practical reasons, including the Egyptians and Babylonians. Around 250 BC the Greek mathematician Archimedes created an algorithm for calculating it. It was approximated to seven digits, using geometrical techniques, in Chinese mathematics, and to about five digits in Indian mathematics in the 5th century AD. The historically first exact formula for π, based on infinite series, was not available until a millennium later, when in the 14th century the Madhava–Leibniz series was discovered in Indian mathematics. In the 20th and 21st centuries, mathematicians and computer scientists discovered new approaches that, when combined with increasing computational power, extended the decimal representation of π to many trillions of digits after the decimal point. Practically all scientific applications require no more than a few hundred digits of π, and many substantially fewer, so the primary motivation for these computations is the quest to find more efficient algorithms for calculating lengthy numeric series, as well as the desire to break records. The extensive calculations involved have also been used to test supercomputers and high-precision multiplication algorithms.
Because its most elementary definition relates to the circle, π is found in many formulae in trigonometry and geometry, especially those concerning circles, ellipses, and spheres. In more modern mathematical analysis, the number is instead defined using the spectral properties of the real number system, as an eigenvalue or a period, without any reference to geometry. It appears therefore in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics, as well as in almost all areas of physics. The ubiquity of π makes it one of the most widely known mathematical constants both inside and outside the scientific community; several books devoted to it have been published, the number is celebrated on Pi Day, and record-setting calculations of the digits of π often result in news headlines. Attempts to memorize the value of π with increasing precision have led to records of over 70,000 digits.
91. Interesting math example #1
92. Interesting math example #2
93. 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 reasons such as rigorous proofs relying on non-elementary techniques, properties, and/or disciplines 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.