Laplacen.

Brit.

/ləˈplɑːs/,

/ləˈplas/,

/laˈplɑːs/,

/laˈplas/, U.S.

/lɑˈplɑs/,

/ləˈplɑs/

Etymology: < the name of Pierre Simon, Marquis de Laplace (1749–1827), French astronomer and mathematician.

1. Philosophy. Used in the possessive to designate an imaginary intelligence described by Laplace which, given the values at any instant of certain physical quantities for all the particles in the universe, could predict in detail the whole of the future from the laws of physics.

ΚΠ

1911 J. Ward Realm of Ends i. 17 The omniscience of Laplace's imaginary spirit with its completed world-formula.

1947 H. Reichenbach Elem. Symbolic Logic viii. 390 Such verification may be technically impossible, although in principle it should be possible to foretell the results of a throw of a die from the initial conditions, given the position of the die, the physiological status of the person considered, and other factors. Let us say that Laplace's superman could do it.

1965 P. Caws Philos. of Sci. xxxix. 300 This intelligence has been called ‘Laplace's demon’, and it has become the patron saint of determinism.

2. Mathematics. Used attributively and in the possessive to designate various concepts and mathematical expressions devised by Laplace or arising out of his work.

a. † Laplace's coefficient n. Obsolete a Legendre polynomial.

ΚΠ

a1830 F. Lunn Elec. in Encycl. Metrop. (1845) IV. 144 If f be the distance of the differential particle dm from the attracted particle V = ∫dm/f. We have now to find the quantity V; this we shall do by expanding it into a series, the coefficients of which have peculiar properties, depending upon a partial differential equation to which they are subject... We shall..distinguish them by that [sc. name] of their illustrious inventor, calling the differential equation and the coefficients..Laplace's equation, and Laplace's coefficients.

1873 J. C. Maxwell Treat. Electr. & Magnetism II. i. ix. 162 The theory of spherical harmonics was first given by Laplace in the third book of his Mécanique Celeste. The harmonics themselves are therefore often called Laplace's Coefficients.

1873 J. C. Maxwell Treat. Electr. & Magnetism II. i. ix. 164 It is shewn in treatises on Laplace's Coefficients that Q_i is the coefficient of hⁱ in the expansion of (1 − 2μh + h²)^−1/2.

b. Laplace's equation n. the equation ∇²V = 0, esp. its representation in Cartesian co-ordinates, ^∂²V/ _∂x² + ^∂²V/ _∂y² + ^∂²V/ _∂z² = 0, where V is a function of x, y, and z.

ΚΠ

1812 Philos. Trans. (Royal Soc.) 102 31 It is exclusively confined to that class of spheroids which, while they differ from spheres, likewise have their radii expressed by rational and integral functions of a point in the surface of a sphere: in this hypothesis Laplace's equation has been rigorously demonstrated.

1813 Philos. Mag. 41 9 In Laplace's equation (d²V/dx²) + (d²V/dy²) + (d²V/dz²) = o,..V is a function of x, y, and z.

1962 D. R. Corson & P. Lorrain Introd. Electromagn. Fields ii. 36 If we introduce into Eq. 2–23 the electrostatic potential V,..∇²V = − ρ/ε₀. This is Poisson's equation. In a region of the field where the charge density ρ is zero, ∇²V = o, which is Laplace's equation.

1962 D. R. Corson & P. Lorrain Introd. Electromagn. Fields iv. 154 Certain cases of symmetry are best treated in spherical polar coordinates. Laplace's equation then takes the form ∇²V = ¹/ _r²^∂/ _∂r r²^∂V/ _∂r + ¹/ _{r² sin θ}^∂/ _∂θ sin θ ^∂V/ _∂θ + ¹/ _{r² sin θ}^∂²V/ _∂ϕ² = 0.

c. Laplace operator n. (also Laplace's operator) = Laplacian n.

ΚΠ

1873 J. C. Maxwell Treat. Electr. & Magnetism I. 29 One of the most remarkable properties of the operator ∇ is that when repeated it becomes ∇² = − (d²/dx² + d²/dy² + d²/dz²), an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator.

1935 L. Pauling & E. B. Wilson Introd. Quantum Mech. iv. 85 This [Schrödinger] equation is often written as ^−h²/ _8π² ∑N_i = 1¹/ _{m_i} ∇_i²Ψ + VΨ = −^h/ _2πi^∂Ψ/ _∂t, in which ∇ i² is the Laplace operator or Laplacian for the ith particle.

1936 P. M. Morse Vibration & Sound vii. 232 We can write the wave equation as ∇²p = (∂²p/∂t²)/c²... The operator ∇² is called the Laplace operator, or the Laplacian: it measures the concentration of a quantity (or, rather, the negative of the concentration). The value of ∇²p at a point is proportional to the difference between the average pressure near a point and the pressure right at the point.

d. Laplace transform n. a function f(x) related to a given function g(t) by the equation f(x) = ∞0 exp (− xt) g(t)dt.

ΚΠ

1944 T. H. Turney Heaviside's Operational Calculus made Easy vii. 84 (heading) The Laplace transform method of circuit analysis.

1962 D. R. Cox Renewal Theory i. 3 One of the main mathematical tools used in renewal theory is the Laplace transform.

e. Laplace transformation n. the transformation by which f(x) is obtained from g(t).

ΚΠ

1949 S. Goldman Transformation Calculus iii. 57 The Laplace transformation transforms f(t), a function of t, into F(s), a function of some new variable s.

1956 Nature 21 Jan. 106/2 The approach is mathematical and based throughout on the use of the Laplace transformation. The book is..intended..as a supplement to introductory text-books on feedback systems and on the functions of a complex variable.

This entry has not yet been fully updated (first published 1976; most recently modified version published online March 2022).

单词	laplace
释义	Laplacen. Brit. /ləˈplɑːs/, /ləˈplas/, /laˈplɑːs/, /laˈplas/, U.S. /lɑˈplɑs/, /ləˈplɑs/ Etymology: < the name of Pierre Simon, Marquis de Laplace (1749–1827), French astronomer and mathematician. 1. Philosophy. Used in the possessive to designate an imaginary intelligence described by Laplace which, given the values at any instant of certain physical quantities for all the particles in the universe, could predict in detail the whole of the future from the laws of physics. ΚΠ 1911 J. Ward Realm of Ends i. 17 The omniscience of Laplace's imaginary spirit with its completed world-formula. 1947 H. Reichenbach Elem. Symbolic Logic viii. 390 Such verification may be technically impossible, although in principle it should be possible to foretell the results of a throw of a die from the initial conditions, given the position of the die, the physiological status of the person considered, and other factors. Let us say that Laplace's superman could do it. 1965 P. Caws Philos. of Sci. xxxix. 300 This intelligence has been called ‘Laplace's demon’, and it has become the patron saint of determinism. 2. Mathematics. Used attributively and in the possessive to designate various concepts and mathematical expressions devised by Laplace or arising out of his work. a. † Laplace's coefficient n. Obsolete a Legendre polynomial. ΚΠ a1830 F. Lunn Elec. in Encycl. Metrop. (1845) IV. 144 If f be the distance of the differential particle dm from the attracted particle V = ∫dm/f. We have now to find the quantity V; this we shall do by expanding it into a series, the coefficients of which have peculiar properties, depending upon a partial differential equation to which they are subject... We shall..distinguish them by that [sc. name] of their illustrious inventor, calling the differential equation and the coefficients..Laplace's equation, and Laplace's coefficients. 1873 J. C. Maxwell Treat. Electr. & Magnetism II. i. ix. 162 The theory of spherical harmonics was first given by Laplace in the third book of his Mécanique Celeste. The harmonics themselves are therefore often called Laplace's Coefficients. 1873 J. C. Maxwell Treat. Electr. & Magnetism II. i. ix. 164 It is shewn in treatises on Laplace's Coefficients that Q_i is the coefficient of hⁱ in the expansion of (1 − 2μh + h²)^−1/2. b. Laplace's equation n. the equation ∇²V = 0, esp. its representation in Cartesian co-ordinates, ^∂²V/ _∂x² + ^∂²V/ _∂y² + ^∂²V/ _∂z² = 0, where V is a function of x, y, and z. ΚΠ 1812 Philos. Trans. (Royal Soc.) 102 31 It is exclusively confined to that class of spheroids which, while they differ from spheres, likewise have their radii expressed by rational and integral functions of a point in the surface of a sphere: in this hypothesis Laplace's equation has been rigorously demonstrated. 1813 Philos. Mag. 41 9 In Laplace's equation (d²V/dx²) + (d²V/dy²) + (d²V/dz²) = o,..V is a function of x, y, and z. 1962 D. R. Corson & P. Lorrain Introd. Electromagn. Fields ii. 36 If we introduce into Eq. 2–23 the electrostatic potential V,..∇²V = − ρ/ε₀. This is Poisson's equation. In a region of the field where the charge density ρ is zero, ∇²V = o, which is Laplace's equation. 1962 D. R. Corson & P. Lorrain Introd. Electromagn. Fields iv. 154 Certain cases of symmetry are best treated in spherical polar coordinates. Laplace's equation then takes the form ∇²V = ¹/ _r²^∂/ _∂r r²^∂V/ _∂r + ¹/ _{r² sin θ}^∂/ _∂θ sin θ ^∂V/ _∂θ + ¹/ _{r² sin θ}^∂²V/ _∂ϕ² = 0. c. Laplace operator n. (also Laplace's operator) = Laplacian n. ΚΠ 1873 J. C. Maxwell Treat. Electr. & Magnetism I. 29 One of the most remarkable properties of the operator ∇ is that when repeated it becomes ∇² = − (d²/dx² + d²/dy² + d²/dz²), an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator. 1935 L. Pauling & E. B. Wilson Introd. Quantum Mech. iv. 85 This [Schrödinger] equation is often written as ^−h²/ _8π² ∑N_i = 1¹/ _{m_i} ∇_i²Ψ + VΨ = −^h/ _2πi^∂Ψ/ _∂t, in which ∇ i² is the Laplace operator or Laplacian for the ith particle. 1936 P. M. Morse Vibration & Sound vii. 232 We can write the wave equation as ∇²p = (∂²p/∂t²)/c²... The operator ∇² is called the Laplace operator, or the Laplacian: it measures the concentration of a quantity (or, rather, the negative of the concentration). The value of ∇²p at a point is proportional to the difference between the average pressure near a point and the pressure right at the point. d. Laplace transform n. a function f(x) related to a given function g(t) by the equation f(x) = ∞0 exp (− xt) g(t)dt. ΚΠ 1944 T. H. Turney Heaviside's Operational Calculus made Easy vii. 84 (heading) The Laplace transform method of circuit analysis. 1962 D. R. Cox Renewal Theory i. 3 One of the main mathematical tools used in renewal theory is the Laplace transform. e. Laplace transformation n. the transformation by which f(x) is obtained from g(t). ΚΠ 1949 S. Goldman Transformation Calculus iii. 57 The Laplace transformation transforms f(t), a function of t, into F(s), a function of some new variable s. 1956 Nature 21 Jan. 106/2 The approach is mathematical and based throughout on the use of the Laplace transformation. The book is..intended..as a supplement to introductory text-books on feedback systems and on the functions of a complex variable. This entry has not yet been fully updated (first published 1976; most recently modified version published online March 2022). < n.1812
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