An Etymological Dictionary of Astronomy and Astrophysics
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1) If a → power series → converges for some nonzero value x₀, then it converges absolutely for any value of x, for which |x| < |x₀|.
2) If a power series → diverges for some nonzero value x₀, then it diverges for any value of x, for which |x| > |x₀|.

Named after the Norwegian mathematician Niels Henrik Abel (1802-1829); → theorem.

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.

A statement of the → conservation of energy in the → steady flow of an → incompressible, → inviscid fluid. Accordingly, the quantity (P/ρ) + gz + (V²/2) 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.

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^kb^{n - k}, where the summation is from k = 0 to n, and (n,k) = n!/[r!(n - k)!]. For n = 2, (a + b)² = a² + 2ab + b². 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).

→ binomial; → theorem.

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

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.

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, X₁, X₂,..., X_n, with a → mean μ and → variance σ², 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.

→ central; → limit; → theorem.

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.

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:
a) the acceleration P would have if it were fixed to the moving system;
b) the acceleration of P with respect to the local moving system; and
c) a compound supplementary → Coriolis acceleration.

→ Coriolis effect; → theorem.

Same as → Gauss's theorem.

→ divergence; → theorem.

Math: A theorem that asserts the existence of at least one object, such as the → solution to a → problem or → equation.

→ existence; → theorem.

In → number theory, the statement that for all → integers, the equation xⁿ + yⁿ = zⁿ 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.

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.

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.

→ gauss; → theorem.

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.

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.

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.

1) If f(x) is a continuous function on the interval from a to b, then:
∫ f(x) dx = f(c)(b - a) (summed from a to b) for at least one point in that interval.
2) More generally, If f(x) and g(x) are continuous functions on the interval from a to b and g(x)≥ 0, then:
∫ f(x)g(x) dx = f(c) ∫ g(x) dx (both integrals summed from a to b).

→ mean; → value; → theorem.

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.

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.

→ Newton; → shell; → theorem.