An Etymological Dictionary of Astronomy and Astrophysics
English-French-Persian

An equation in which the → dependent variable y and all its differential coefficients occur only in the first degree. A linear differential equation of → order  order n has the form:
f_n(x)y⁽ⁿ⁾ + f_n-1(x)y^(n-1) + ... + f₁(x)y^' + f₀(x)y = Q(x),
where f₀(x), f₁(x), ..., f_n(x) and Q(x) are each continuous functions of x defined on a common interval I and f_n(x)≠ 0 in I. A linear differential equation cannot have, for example, terms such as y² or (y^')^1/2. See also: → homogeneous linear differential equation; → nonhomogeneous linear differential equation.

→ linear; → differential; → equation.

An equation composed of first degree variables and representing a straight line.

→ linear; → equation.

A differential equation that has been derived from an original nonlinear equation.

Linearized, p.p. of → linearize; → differential; → equation.

A set of four fundamental equations that describe the electric and magnetic fields arising from varying electric charges and magnetic fields, electric currents, charge distributions, and how those fields change in time. In their vector differential form, these equations are:
i) ∇.E = ρ/ε₀ (→ Gauss's law for electricity),
ii) ∇.B = 0 (→ Gauss's law for magnetism),
iii) ∇ x E = -∂B/∂t (→ Faraday's law of induction),
iv) ∇ x B = μ₀J + μ₀ε₀∂E/∂t (→ Ampere's law), with c² = 1/(μ₀ε₀), where E is → electric intensity, B is → magnetic flux density, ρ is → charge density, ε₀ is → permittivity, μ₀ is → permeability, J is → current density, and c is → speed of light.

→ maxwell. It should be emphasized that the equations originally published by James Clerk Maxwell in 1873 (in A Treatise on Electricity and Magnetism) were 20 in number, had 20 variables, and were in scalar form. The German physicist Heinrich Rudolf Hertz (1857-1894) reduced them to 12 scalar equations (1884). It was the English mathematician/physicist Oliver Heaviside (1850-1925) who expressed Maxwell's equations in vector form using the notations of → gradient, → divergence, and → curl of a vector, thus simplifying them to the present 4 equations (1886). Before Einstein these equations were known as Maxwell-Heaviside-Hertz equations, Einstein (1940) popularized the name "Maxwell's Equations;" → equation.

One of a set of → differential equations that describes the motion of a → fluid as a function of → pressure, → density, total external force, and → viscosity. See also → Euler equation.

Named after Claude-Louis Navier (1785-1836), a French engineer and physicist, and George Gabriel Stokes, → stokes; → equation.

In → geometric optics, an expression relating the → focal lengths of an → optical system (f and f') and the object x and image x' distances measured from the respective focal points. Thus, ff' = xx'. Same as Newton's formula.

→ Newton; → equation.

An equation that is independent of the units of measurement as it only involves nondimensional numbers, parameters, and variables.

→ non-; → dimensional; → equation.

A → linear differential equation if Q(x)≠ 0 on interval I.

→ nonhomogeneous; → linear; → differential; → equation.

A → differential equation in which the unknown function depends on only one → independent variable, as contrasted with a → partial differential equation.

→ ordinary; → differential; → equation.

Any of a set of equations that defines the coordinates of the dependent variables of a curve or surface in terms of one or more independent variables or parameters.

→ parametric; → equation.

A type of differential equation involving an unknown function (or functions) of several independent variables and its (or their) partial derivatives with respect to those variables.

→ partial; → differential; → equation.

A systematic observational error due to the characteristics of the observer.

Personal, adj. of → person; → equation.

Any equation governing the behavior of a → perturbation.

→ perturbation; → equation.

An equation (∇²φ = 4πGρ) which relates the gravitational (or electromagnetic) potential to the mass density (or charge density).

→ Poisson distribution; → equation.

An equation for a curve written in terms of the → polar coordinates.

→ polar; → equation.

An equation of the form a₀ + a₁x + a₂x² + ... + a_nxⁿ, where a₀ ... a_n are → real numbers and a_n≠ 0. Same as → algebraic equation.

→ polynomial; → equation.

An equation with the general form of ax² + bx +c = 0, in which the highest power of the unknown is the second power (square).

→ quadratic; → equation.

An equation containing unknowns of the fourth power; the general form: ax⁴ + bx³ + cx² + dx + e = 0.

From L. quart(us) "fourth" (→ quarter) + → -ic; → equation.

hamugeš, → equation; câromân, from cârom "fourth," from câr, cahâr "four" + -om "-th" + -ik, → -ic.

An equation containing unknowns of the fifth power.

→ quintic; → equation.

→ radiative transfer equation.

→ radiation; → transfer; → equation.