In mathematics, a law is a formula that is always true within a given context. [1] Laws describe a relationship, between two or more expressions or terms (which may contain variables), usually using equality or inequality, [2] or between formulas themselves, for instance, in mathematical logic. For example, the formula is true for all real numbers a, and is therefore a law. Laws over an equality are called identities. [3] For example, and are identities. [4] Mathematical laws are distinguished from scientific laws which are based on observations, and try to describe or predict a range of natural phenomena. [5] The more significant laws are often called theorems.
Triangle inequality: If a, b, and c are the lengths of the sides of a triangle then the triangle inequality states that
with equality only in the degenerate case of a triangle with zero area. In Euclidean geometry and some other geometries, the triangle inequality is a theorem about vectors and vector lengths (norms):
where the length of the third side has been replaced by the length of the vector sum u + v. When u and v are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values.
Pythagorean theorem: It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: [6]
Geometrically, trigonometric identities are identities involving certain functions of one or more angles. [7] They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
One of the most prominent examples of trigonometric identities involves the equation which is true for all real values of . On the other hand, the equation
is only true for certain values of , not all. For example, this equation is true when but false when .
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity , the addition formula for ), which can be used to break down expressions of larger angles into those with smaller constituents.
Cauchy–Schwarz inequality: An upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. [8]
The Cauchy–Schwarz inequality states that for all vectors and of an inner product space
where is the inner product. Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean norm, called the canonical or inducednorm, where the norm of a vector is denoted and defined by
where is always a non-negative real number (even if the inner product is complex-valued). By taking the square root of both sides of the above inequality, the Cauchy–Schwarz inequality can be written in its more familiar form in terms of the norm: [9] [10]
Moreover, the two sides are equal if and only if and are linearly dependent. [11] [12] [13]
Pigeonhole principle: If n items are put into m containers, with n > m, then at least one container must contain more than one item. [14] For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into.
De Morgan's laws: In propositional logic and Boolean algebra, De Morgan's laws, [15] [16] [17] also known as De Morgan's theorem, [18] are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathematician. The rules allow the expression of conjunctions and disjunctions purely in terms of each other via negation. The rules can be expressed in English as:
The three Laws of thought are:
Benford's law is an observation that in many real-life sets of numerical data, the leading digit is likely to be small. [21] In sets that obey the law, the number 1 appears as the leading significant digit about 30% of the time, while 9 appears as the leading significant digit less than 5% of the time. Uniformly distributed digits would each occur about 11.1% of the time. [22]
Strong law of small numbers, in a humorous way, states any given small number appears in far more contexts than may seem reasonable, leading to many apparently surprising coincidences in mathematics, simply because small numbers appear so often and yet are so few.
...there is no doubt that this is one of the most widely used and most important inequalities in all of mathematics.
Equality holds iff <c|c>=0 or |c>=0. From the definition of |c>, we conclude that |a> and |b> must be proportional.
This inequality is an equality if and only if one of u, v is a scalar multiple of the other.
In mathematics, the trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The Cauchy–Schwarz inequality is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics.
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers, and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used. It is often called the inner product of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space.
In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.
In mathematics, particularly in linear algebra, a skew-symmetricmatrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three dimensional space. Specifically, they encode information about an axis-angle rotation about an arbitrary axis. Rotation and orientation quaternions have applications in computer graphics, computer vision, robotics, navigation, molecular dynamics, flight dynamics, orbital mechanics of satellites, and crystallographic texture analysis.
In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an complex matrix is an matrix obtained by transposing and applying complex conjugation to each entry. There are several notations, such as or , , or .
In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same value for all values of the variables within a certain domain of discourse. In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined. For example, and are identities. Identities are sometimes indicated by the triple bar symbol ≡ instead of =, the equals sign. Formally, an identity is a universally quantified equality.
In quantum mechanics and computing, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch.
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix
The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1851, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the system. They can be determined from directly observable phenomena. The original Stokes paper was discovered independently by Francis Perrin in 1942 and by Subrahamanyan Chandrasekhar in 1947, who named it as the Stokes parameters.
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The polarization identity shows that a norm can arise from at most one inner product; however, there exist norms that do not arise from any inner product.
In spherical trigonometry, the law of cosines is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.
Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.
In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an optical system. For any such system, a plot of the quadratures against each other, possibly as functions of time, is called a phase diagram. If the quadratures are functions of time then the optical phase diagram can show the evolution of a quantum optical system with time.
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
The following are important identities in vector algebra. Identities that only involve the magnitude of a vector and the dot product of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product A×B only apply in three dimensions, since the cross product is only defined there. Most of these relations can be dated to founder of vector calculus Josiah Willard Gibbs, if not earlier.
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator.
The concept of angles between lines, between two planes or between a line and a plane can be generalized to arbitrary dimensions. This generalization was first discussed by Camille Jordan. For any pair of flats in a Euclidean space of arbitrary dimension one can define a set of mutual angles which are invariant under isometric transformation of the Euclidean space. If the flats do not intersect, their shortest distance is one more invariant. These angles are called canonical or principal. The concept of angles can be generalized to pairs of flats in a finite-dimensional inner product space over the complex numbers.