Abel's theorem farbin-e Abel Fr.: théorème d'Abel 1) If a → power series → converges
for some nonzero value
x_{0}, then it converges absolutely for any value of x, for
which |x| < |x_{0}|. Named after the Norwegian mathematician Niels Henrik Abel (1802-1829); → theorem. |
Bayes' theorem farbin-e Bayes Fr.: théorème de Bayes A theorem in probability theory concerned with determining the → conditional probability of an event when another event has occurred. Bayes' theorem allows revision of the original probability with new information. Its simplest form is: P(A|B) = P(B|A) P(A)/P(B), where P(A): independent probability of A, also called prior probability; P(B): independent probability of B; P(B|A): conditional probability of B given A has occurred; P(A|B): conditional probability of A given B has occurred, also called posterior probability. Same as Bayes' rule. Named after its proponent, the British mathematician Reverend Thomas Bayes (1702-1761). However, Bayes did not publish the theorem during his lifetime; instead, it was presented two years after his death to the Royal Society of London. |
Bernoulli's theorem farbin-e Bernoulli Fr.: théorème de Bernoulli A statement of the → conservation of energy in the → steady flow of an → incompressible, → inviscid fluid. Accordingly, the quantity (P/ρ) + gz + (V^{2}/2) is → constant along any → streamline, where P is the fluid → pressure, V is the fluid → velocity, ρ is the mass → density of the fluid, g is the acceleration due to → gravity, and z is the vertical → height. This equation affirms that if the internal velocity of the flow goes up, the internal pressure must drop. Therefore, the flow becomes more constricted if the velocity field within it increases. Same as the → Bernoulli equation. After Daniel Bernoulli (1700-1782), the Swiss physicist and mathematician who put forward the theorem in his book Hydrodynamica in 1738; → theorem. |
binomial theorem farbin-e donâmin Fr.: théorème du binôme A rule for writing an equivalent expansion of an expression such as (a + b)^{n} without having to perform all multiplications involved. → binomial expansion. The general expression is (a + b)^{n} = &Sigma (n,k)a^{k}b^{n - k}, where the summation is from k = 0 to n, and (n,k) = n!/[r!(n - k)!]. For n = 2, (a + b)^{2} = a^{2} + 2ab + b^{2}. Historically, the binomial theorem as applied to (a + b)^{2} was known to Euclid (320 B.C.) and other early Greek mathematicians. In the tenth century the Iranian mathematician Karaji (953-1029) knew the binomial theorem and its accompanying table of → binomial coefficients, now known as → Pascal's triangle. Subsequently Omar Khayyam (1048-1131) asserted that he could find the 4th, 5th, 6th, and higher roots of numbers by a special law which did not depend on geometric figures. Khayyam's treatise concerned with his findings is lost. In China there appeared in 1303 a work containing the binomial coefficients arranged in triangular form. The complete generalization of the binomial theorem for all values of n, including negative integers, was established by Isaac Newton (1642-1727). |
Birkhoff's theorem farbin-e Birkhoff Fr.: théorème de Birkhoff For a four dimensional → space-time, the → Schwarzschild metric is the only solution of → Einstein's field equations which describes the gravitational field created by a spherically symmetrical distribution of mass. The theorem implies that the gravitational field outside a sphere is necessarily static, and that the metric inside a spherical shell of matter is necessarily flat. The theorem was first demonstrated in 1923 by George David Birkhoff (1884-1944), an American mathematician; → theorem |
Cauchy's theorem farbin-e Cauchy Fr.: théorème de Cauchy If f(x) and φ(x) are two → continuous functions on the → interval [a,b] and → differentiable within it, and φ'(x) does not vanish anywhere inside the interval, there will be found, in [a,b], some point x = c, such that [f(b) - f(a)] / [φ(b) - φ(a)] = f'(c) / φ'(c). → Cauchy's equation; → theorem. |
central limit theorem farbin-e hadd-e markazi Fr.: théorème central limite A statement about the characteristics of the sampling distribution of means of → random samples from a given → statistical population. For any set of independent, identically distributed random variables, X_{1}, X_{2},..., X_{n}, with a → mean μ and → variance σ^{2}, the distribution of the means is equal to the mean of the population from which the samples were drawn. Moreover, if the original population has a → normal distribution, the sampling distribution of means will also be normal. If the original population is not normally distributed, the sampling distribution of means will increasingly approximate a normal distribution as sample size increases. |
convolution theorem farbin-e hamâgiš Fr.: théorème de convolution A theorem stating that the → Fourier transform of the convolution of f(x) and g(x) is equal to the product of the Fourier transform of f(x) and g(x): F{f*g} = F{f}.F{g}. → convolution; → theorem. |
Coriolis theorem farbin-e Coriolis Fr.: théorème de Coriolis The → absolute acceleration of a point P, which is moving
with respect to a local → reference frame
that is also in motion, is equal to the vector
sum of: → Coriolis effect; → theorem. |
divergence theorem farbin-e vâgerâyi Fr.: théorème de flux-divergence Same as → Gauss's theorem. → divergence; → theorem. |
existence theorem farbin-e hastumandi, ~ hasti Fr.: théorème d'existence Math: A theorem that asserts the existence of at least one object, such as the → solution to a → problem or → equation. |
Fermat's last theorem vâpasin farbin-e Fermat Fr.: dernier théorème de Fermat In → number theory, the statement that for all → integers, the equation x^{n} + y^{n} = z^{n} has no solution in → positive integer. After 358 years of effort by mathematicians to prove the theorem, a complete proof was found by Andrew Wiles in 1995. → Fermat's principle; → last; → theorem. |
Fourier theorem farbin-e Fourier Fr.: théorème de Fourier Any finite periodic motion may be analyzed into components, each of which is a simple harmonic motion of definite and determinable amplitudes and phase. → Fourier analysis; → theorem. |
Gauss's theorem farbin-e Gauss Fr.: théorème de Gauss The total normal induction over any closed surface drawn in an electric field is equal to 4π times the total charge of electricity inside the closed surface. Gauss's theorem applies also to other vector fields such as magnetic, gravitational, and fluid velocity fields. The theorem can more generally be stated as: the total flux of a vector field through a closed surface is equal to the volume → integral of the vector taken over the enclosed volume. Also known as → divergence theorem, Ostrogradsky's theorem, and Gauss-Ostrogradsky theorem. |
Helmholtz's theorem farbin-e Helmholtz Fr.: théorème de Helmholtz A → decomposition theorem, whereby a continuous → vector field, F, can be broken down into the sum of a → gradient and a → curl term: F = -∇φ + ∇ xA, where φ is called the → scalar potential and A the → vector potential. → Helmholtz free energy; → theorem. |
Larmor's theorem farbin-e Larmor Fr.: théorème de Larmor If a system of → charged particles, all having the same ratio of charge to mass (q/m), acted on by their mutual forces, and by a central force toward a common center, is subject in addition to a weak uniform magnetic field (B), its possible motions will be the same as the motions it could perform without the magnetic field, superposed upon a slow → precession of the entire system about the center of force with angular velocity ω = -(q/2mc)B. → Larmor frequency; → theorem. |
Liouville's theorem farbin-e Liouville Fr.: théorème de Liouville A key theorem in statistical mechanics of classical systems which states that the motion of phase-space points defined by Hamilton's equations conserves phase-space volume. After Joseph Liouville (1809-1882), a French mathematician; → theorem. |
mean value theorem farbin-e arzeš-e miyângin Fr.: théorème des accroissements finis 1) If f(x) is a continuous function on the interval from a to b, then: |
Nernst heat theorem farbin-e garmâ-ye Nernst Fr.: théorème de Nernst The entropy change for chemical reactions involving crystalline solid is zero at the absolute zero of temperature. Also known as the third law of thermodynamics. → Nernst effect; → heat; → theorem. |
Newton's shell theorem farbin-e puste-ye Newton Fr.: théorème de Newton In classical mechanics, an analytical method applied to a material sphere to determine the gravitational field at a point outside or inside the sphere. Newton's shell theorem states that: 1) The gravitational field outside a uniform spherical shell (i.e. a hollow ball) is the same as if the entire mass of the shell is concentrated at the center of the sphere. 2) The gravitational field inside the spherical shell is zero, regardless of the location within the shell. 3) Inside a solid sphere of constant density, the gravitational force varies linearly with distance from the center, being zero at the center of mass. For the relativistic generalization of this theorem, see → Birkhoff's theorem. |