Misplaced Pages

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131-511: The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics : the Laplacian of the gravitational potential due to a given mass density distribution is a constant multiple of that density distribution. Solutions of Laplace's equation Δ f = 0 are called harmonic functions and represent

262-506: A + h e i ) − f ( a ) h . {\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a} )&=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i-1},a_{i}+h,a_{i+1}\,\ldots ,a_{n})\ -f(a_{1},\ldots ,a_{i},\dots ,a_{n})}{h}}\\&=\lim _{h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e_{i}} )-f(\mathbf {a} )}{h}}\,.\end{aligned}}} Where e i {\displaystyle \mathbf {e_{i}} }

393-402: A 1 , … , a i − 1 , a i + h , a i + 1 … , a n )   − f ( a 1 , … , a i , … , a n ) h = lim h → 0 f (

524-419: A ) ) . {\displaystyle \nabla f(a)=\left({\frac {\partial f}{\partial x_{1}}}(a),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a)\right).} This vector is called the gradient of f at a . If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇ f which takes the point a to the vector ∇ f ( a ) . Consequently, the gradient produces

655-533: A , x sin ⁡ θ + y cos ⁡ θ + b ) ) = ( Δ f ) ( x cos ⁡ θ − y sin ⁡ θ + a , x sin ⁡ θ + y cos ⁡ θ + b ) {\displaystyle \Delta (f(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b))=(\Delta f)(x\cos \theta -y\sin \theta +a,x\sin \theta +y\cos \theta +b)} for all θ ,

786-416: A barotropic two-dimensional sheet flow. Coriolis effects are introduced as well as lateral forcing by gravity. Laplace obtained these equations by simplifying the fluid dynamic equations. But they can also be derived from energy integrals via Lagrange's equation . For a fluid sheet of average thickness D , the vertical tidal elevation ζ , as well as the horizontal velocity components u and v (in

917-440: A plane have polar coordinates ( r , θ) and ( r ', θ'), where r ' ≥ r , then, by elementary manipulation, the reciprocal of the distance between the points, d , can be written as: This expression can be expanded in powers of r / r ' using Newton's generalised binomial theorem to give: The sequence of functions P k (cos φ) is the set of so-called "associated Legendre functions" and their usefulness arises from

1048-392: A scalar function f ( x ) = f ( x 1 , x 2 , … , x n ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{n})} along a vector v = ( v 1 , … , v n ) {\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n})}

1179-2472: A vector field . A common abuse of notation is to define the del operator ( ∇ ) as follows in three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} with unit vectors i ^ , j ^ , k ^ {\displaystyle {\hat {\mathbf {i} }},{\hat {\mathbf {j} }},{\hat {\mathbf {k} }}} : ∇ = [ ∂ ∂ x ] i ^ + [ ∂ ∂ y ] j ^ + [ ∂ ∂ z ] k ^ {\displaystyle \nabla =\left[{\frac {\partial }{\partial x}}\right]{\hat {\mathbf {i} }}+\left[{\frac {\partial }{\partial y}}\right]{\hat {\mathbf {j} }}+\left[{\frac {\partial }{\partial z}}\right]{\hat {\mathbf {k} }}} Or, more generally, for n -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} with coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} and unit vectors e ^ 1 , … , e ^ n {\displaystyle {\hat {\mathbf {e} }}_{1},\ldots ,{\hat {\mathbf {e} }}_{n}} : ∇ = ∑ j = 1 n [ ∂ ∂ x j ] e ^ j = [ ∂ ∂ x 1 ] e ^ 1 + [ ∂ ∂ x 2 ] e ^ 2 + ⋯ + [ ∂ ∂ x n ] e ^ n {\displaystyle \nabla =\sum _{j=1}^{n}\left[{\frac {\partial }{\partial x_{j}}}\right]{\hat {\mathbf {e} }}_{j}=\left[{\frac {\partial }{\partial x_{1}}}\right]{\hat {\mathbf {e} }}_{1}+\left[{\frac {\partial }{\partial x_{2}}}\right]{\hat {\mathbf {e} }}_{2}+\dots +\left[{\frac {\partial }{\partial x_{n}}}\right]{\hat {\mathbf {e} }}_{n}} The directional derivative of

1310-482: A , and b . In arbitrary dimensions, Δ ( f ∘ ρ ) = ( Δ f ) ∘ ρ {\displaystyle \Delta (f\circ \rho )=(\Delta f)\circ \rho } whenever ρ is a rotation, and likewise: Δ ( f ∘ τ ) = ( Δ f ) ∘ τ {\displaystyle \Delta (f\circ \tau )=(\Delta f)\circ \tau } whenever τ

1441-427: A bounded domain. When Ω is the n -sphere , the eigenfunctions of the Laplacian are the spherical harmonics . The vector Laplace operator , also denoted by ∇ 2 {\displaystyle \nabla ^{2}} , is a differential operator defined over a vector field . The vector Laplacian is similar to the scalar Laplacian; whereas the scalar Laplacian applies to a scalar field and returns

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1572-671: A case, evaluation of the function must be expressed in an unwieldy manner as ∂ f ( x , y , z ) ∂ x ( 17 , u + v , v 2 ) {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}(17,u+v,v^{2})} or ∂ f ( x , y , z ) ∂ x | ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle \left.{\frac {\partial f(x,y,z)}{\partial x}}\right|_{(x,y,z)=(17,u+v,v^{2})}} in order to use

1703-407: A chemical concentration, then the net flux of u through the boundary ∂ V (also called S ) of any smooth region V is zero, provided there is no source or sink within V : ∫ S ∇ u ⋅ n d S = 0 , {\displaystyle \int _{S}\nabla u\cdot \mathbf {n} \,dS=0,} where n is the outward unit normal to

1834-505: A consequence, the spherical Laplacian of a function defined on S ⊂ R can be computed as the ordinary Laplacian of the function extended to R ∖{0} so that it is constant along rays, i.e., homogeneous of degree zero. The Laplacian is invariant under all Euclidean transformations : rotations and translations . In two dimensions, for example, this means that: Δ ( f ( x cos ⁡ θ − y sin ⁡ θ +

1965-399: A derivation of Kepler's laws , which describe the motion of the planets, from his laws of motion and his law of universal gravitation . However, though Newton had privately developed the methods of calculus, all his published work used cumbersome geometric reasoning, unsuitable to account for the more subtle higher-order effects of interactions between the planets. Newton himself had doubted

2096-515: A few centimeters. Measurements from the CHAMP satellite closely match the models based on the TOPEX data. Accurate models of tides worldwide are essential for research since the variations due to tides must be removed from measurements when calculating gravity and changes in sea levels. In 1776, Laplace formulated a single set of linear partial differential equations , for tidal flow described as

2227-467: A function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x} is variously denoted by It can be thought of as the rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} ,

2358-568: A function of several variables is the case of a scalar-valued function f ( x 1 , … , x n ) {\displaystyle f(x_{1},\ldots ,x_{n})} on a domain in Euclidean space R n {\displaystyle \mathbb {R} ^{n}} (e.g., on R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). In this case f has

2489-509: A function. The partial derivative of f at the point a = ( a 1 , … , a n ) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots ,a_{n})\in U} with respect to the i -th variable x i is defined as ∂ ∂ x i f ( a ) = lim h → 0 f (

2620-545: A memoir in which he first explored the possible influences of a purported luminiferous ether or of a law of gravitation that did not act instantaneously. He ultimately returned to an intellectual investment in Newtonian gravity. Euler and Lagrange had made a practical approximation by ignoring small terms in the equations of motion. Laplace noted that though the terms themselves were small, when integrated over time they could become important. Laplace carried his analysis into

2751-479: A partial derivative ∂ f / ∂ x j {\displaystyle \partial f/\partial x_{j}} with respect to each variable x j . At the point a , these partial derivatives define the vector ∇ f ( a ) = ( ∂ f ∂ x 1 ( a ) , … , ∂ f ∂ x n (

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2882-592: A place in astronomical tables." The result is embodied in the Exposition du système du monde and the Mécanique céleste . The former was published in 1796, and gives a general explanation of the phenomena, but omits all details. It contains a summary of the history of astronomy. This summary procured for its author the honour of admission to the forty of the French Academy and is commonly esteemed one of

3013-465: A potential occurs in fluid dynamics , electromagnetism and other areas. Rouse Ball speculated that it might be seen as "the outward sign" of one of the a priori forms in Kant's theory of perception . The spherical harmonics turn out to be critical to practical solutions of Laplace's equation. Laplace's equation in spherical coordinates , such as are used for mapping the sky, can be simplified, using

3144-412: A prize to Fresnel for his new approach. Using corpuscular theory, Laplace also came close to propounding the concept of the black hole . He suggested that gravity could influence light and that there could be massive stars whose gravity is so great that not even light could escape from their surface (see escape velocity ). However, this insight was so far ahead of its time that it played no role in

3275-745: A scalar quantity, the vector Laplacian applies to a vector field , returning a vector quantity. When computed in orthonormal Cartesian coordinates , the returned vector field is equal to the vector field of the scalar Laplacian applied to each vector component. The vector Laplacian of a vector field A {\displaystyle \mathbf {A} } is defined as ∇ 2 A = ∇ ( ∇ ⋅ A ) − ∇ × ( ∇ × A ) . {\displaystyle \nabla ^{2}\mathbf {A} =\nabla (\nabla \cdot \mathbf {A} )-\nabla \times (\nabla \times \mathbf {A} ).} This definition can be seen as

3406-437: A sense made precise by the diffusion equation . This interpretation of the Laplacian is also explained by the following fact about averages. Given a twice continuously differentiable function f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } and a point p ∈ R n {\displaystyle p\in \mathbb {R} ^{n}} ,

3537-475: A similar problem though he was using Newtonian-type geometric reasoning. Laplace described Clairaut's work as being "in the class of the most beautiful mathematical productions". However, Rouse Ball alleges that the idea "was appropriated from Joseph Louis Lagrange , who had used it in his memoirs of 1773, 1777 and 1780". The term "potential" itself was due to Daniel Bernoulli , who introduced it in his 1738 memoire Hydrodynamica . However, according to Rouse Ball,

3668-992: A smooth function, and let K ⊂ Ω {\displaystyle K\subset \Omega } be a connected compact set. If u {\displaystyle u} is superharmonic, then, for every x ∈ K {\displaystyle x\in K} , we have u ( x ) ≥ inf Ω u + c ‖ u ‖ L 1 ( K ) , {\displaystyle u(x)\geq \inf _{\Omega }u+c\lVert u\rVert _{L^{1}(K)}\;,} for some constant c > 0 {\displaystyle c>0} depending on Ω {\displaystyle \Omega } and K {\displaystyle K} . Pierre-Simon de Laplace Pierre-Simon, Marquis de Laplace ( / l ə ˈ p l ɑː s / ; French: [pjɛʁ simɔ̃ laplas] ; 23 March 1749 – 5 March 1827)

3799-603: Is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C function. This can be used to generalize for vector valued functions, f : U → R m {\displaystyle f:U\to \mathbb {R} ^{m}} , by carefully using a componentwise argument. The partial derivative ∂ f ∂ x {\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined on U and can again be partially differentiated. If

3930-734: Is a coordinate dependent result, and is not general. An example of the usage of the vector Laplacian is the Navier-Stokes equations for a Newtonian incompressible flow : ρ ( ∂ v ∂ t + ( v ⋅ ∇ ) v ) = ρ f − ∇ p + μ ( ∇ 2 v ) , {\displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} \right)=\rho \mathbf {f} -\nabla p+\mu \left(\nabla ^{2}\mathbf {v} \right),} where

4061-572: Is a corresponding eigenfunction f with: − Δ f = λ f . {\displaystyle -\Delta f=\lambda f.} This is known as the Helmholtz equation . If Ω is a bounded domain in R , then the eigenfunctions of the Laplacian are an orthonormal basis for the Hilbert space L (Ω) . This result essentially follows from the spectral theorem on compact self-adjoint operators , applied to

Laplace operator - Misplaced Pages Continue

4192-460: Is a function of more than one variable. For instance, z = f ( x , y ) = x 2 + x y + y 2 . {\displaystyle z=f(x,y)=x^{2}+xy+y^{2}.} The graph of this function defines a surface in Euclidean space . To every point on this surface, there are an infinite number of tangent lines . Partial differentiation

4323-474: Is a translation. (More generally, this remains true when ρ is an orthogonal transformation such as a reflection .) In fact, the algebra of all scalar linear differential operators, with constant coefficients, that commute with all Euclidean transformations, is the polynomial algebra generated by the Laplace operator. The spectrum of the Laplace operator consists of all eigenvalues λ for which there

4454-598: Is any smooth region with boundary ∂ V , then by Gauss's law the flux of the electrostatic field E across the boundary is proportional to the charge enclosed: ∫ ∂ V E ⋅ n d S = ∫ V div ⁡ E d V = 1 ε 0 ∫ V q d V . {\displaystyle \int _{\partial V}\mathbf {E} \cdot \mathbf {n} \,dS=\int _{V}\operatorname {div} \mathbf {E} \,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.} where

4585-536: Is by Marquis de Condorcet from 1770, who used it for partial differences . The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841. Like ordinary derivatives, the partial derivative is defined as a limit . Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f : U → R {\displaystyle f:U\to \mathbb {R} }

4716-476: Is defined as the divergence of the gradient of the tensor: ∇ 2 T = ( ∇ ⋅ ∇ ) T . {\displaystyle \nabla ^{2}\mathbf {T} =(\nabla \cdot \nabla )\mathbf {T} .} For the special case where T {\displaystyle \mathbf {T} } is a scalar (a tensor of degree zero), the Laplacian takes on

4847-522: Is mainly historical, but it gives as appendices the results of Laplace's latest researches. The Mécanique céleste contains numerous of Laplace's own investigations but many results are appropriated from other writers with little or no acknowledgement. The volume's conclusions, which are described by historians as the organised result of a century of work by other writers as well as Laplace, are presented by Laplace if they were his discoveries alone. Jean-Baptiste Biot , who assisted Laplace in revising it for

4978-607: Is the D'Alembertian , used in the Klein–Gordon equation . First of all, we say that a smooth function u : Ω ⊂ R N → R {\displaystyle u\colon \Omega \subset \mathbb {R} ^{N}\to \mathbb {R} } is superharmonic whenever − Δ u ≥ 0 {\displaystyle -\Delta u\geq 0} . Let u : Ω → R {\displaystyle u\colon \Omega \to \mathbb {R} } be

5109-570: Is the Laplace–Beltrami operator on the ( N − 1) -sphere, known as the spherical Laplacian. The two radial derivative terms can be equivalently rewritten as: 1 r N − 1 ∂ ∂ r ( r N − 1 ∂ f ∂ r ) . {\displaystyle {\frac {1}{r^{N-1}}}{\frac {\partial }{\partial r}}\left(r^{N-1}{\frac {\partial f}{\partial r}}\right).} As

5240-518: Is the function ∇ v f {\displaystyle \nabla _{\mathbf {v} }{f}} defined by the limit ∇ v f ( x ) = lim h → 0 f ( x + h v ) − f ( x ) h . {\displaystyle \nabla _{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\to 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.} Suppose that f

5371-402: Is the unit vector of i -th variable x i . Even if all partial derivatives ∂ f / ∂ x i ( a ) {\displaystyle \partial f/\partial x_{i}(a)} exist at a given point a , the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f

Laplace operator - Misplaced Pages Continue

5502-422: Is the Laplace operator, and the entire equation Δ u = 0 is known as Laplace's equation . Solutions of the Laplace equation, i.e. functions whose Laplacian is identically zero, thus represent possible equilibrium densities under diffusion. The Laplace operator itself has a physical interpretation for non-equilibrium diffusion as the extent to which a point represents a source or sink of chemical concentration, in

5633-434: Is the act of choosing one of these lines and finding its slope . Usually, the lines of most interest are those that are parallel to the xz -plane, and those that are parallel to the yz -plane (which result from holding either y or x constant, respectively). To find the slope of the line tangent to the function at P (1, 1) and parallel to the xz -plane, we treat y as a constant. The graph and this plane are shown on

5764-470: Is the dimension of the space, f s h e l l R {\displaystyle f_{shell_{R}}} is the average value of f {\displaystyle f} on the surface of a n-sphere of radius R, ∫ s h e l l R f ( r → ) d r n − 1 {\displaystyle \int _{shell_{R}}f({\overrightarrow {r}})dr^{n-1}}

5895-407: Is the surface integral over a n-sphere of radius R, and A n − 1 {\displaystyle A_{n-1}} is the hypervolume of the boundary of a unit n-sphere . In the physical theory of diffusion , the Laplace operator arises naturally in the mathematical description of equilibrium . Specifically, if u is the density at equilibrium of some quantity such as

6026-475: The Académie . Laplace seems to have regarded analysis merely as a means of attacking physical problems, though the ability with which he invented the necessary analysis is almost phenomenal. As long as his results were true he took but little trouble to explain the steps by which he arrived at them; he never studied elegance or symmetry in his processes, and it was sufficient for him if he could by any means solve

6157-520: The Bayesian interpretation of probability was developed mainly by Laplace. Laplace formulated Laplace's equation , and pioneered the Laplace transform which appears in many branches of mathematical physics , a field that he took a leading role in forming. The Laplacian differential operator , widely used in mathematics, is also named after him. He restated and developed the nebular hypothesis of

6288-465: The Dirichlet energy functional stationary : E ( f ) = 1 2 ∫ U ‖ ∇ f ‖ 2 d x . {\displaystyle E(f)={\frac {1}{2}}\int _{U}\lVert \nabla f\rVert ^{2}\,dx.} To see this, suppose f  : U → R is a function, and u  : U → R is a function that vanishes on

6419-643: The Helmholtz decomposition of the vector Laplacian. In Cartesian coordinates , this reduces to the much simpler form as ∇ 2 A = ( ∇ 2 A x , ∇ 2 A y , ∇ 2 A z ) , {\displaystyle \nabla ^{2}\mathbf {A} =(\nabla ^{2}A_{x},\nabla ^{2}A_{y},\nabla ^{2}A_{z}),} where A x {\displaystyle A_{x}} , A y {\displaystyle A_{y}} , and A z {\displaystyle A_{z}} are

6550-767: The Voss - Weyl formula for the divergence . In spherical coordinates in N dimensions , with the parametrization x = rθ ∈ R with r representing a positive real radius and θ an element of the unit sphere S , Δ f = ∂ 2 f ∂ r 2 + N − 1 r ∂ f ∂ r + 1 r 2 Δ S N − 1 f {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {N-1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}\Delta _{S^{N-1}}f} where Δ S

6681-1125: The azimuthal angle and θ the zenith angle or co-latitude . In general curvilinear coordinates ( ξ , ξ , ξ ): Δ = ∇ ξ m ⋅ ∇ ξ n ∂ 2 ∂ ξ m ∂ ξ n + ∇ 2 ξ m ∂ ∂ ξ m = g m n ( ∂ 2 ∂ ξ m ∂ ξ n − Γ m n l ∂ ∂ ξ l ) , {\displaystyle \Delta =\nabla \xi ^{m}\cdot \nabla \xi ^{n}{\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}+\nabla ^{2}\xi ^{m}{\frac {\partial }{\partial \xi ^{m}}}=g^{mn}\left({\frac {\partial ^{2}}{\partial \xi ^{m}\,\partial \xi ^{n}}}-\Gamma _{mn}^{l}{\frac {\partial }{\partial \xi ^{l}}}\right),} where summation over

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6812-445: The curl to find an equation for vorticity . Under certain conditions this can be further rewritten as a conservation of vorticity. During the years 1784–1787 he published some papers of exceptional power. Prominent among these is one read in 1783, reprinted as Part II of Théorie du Mouvement et de la figure elliptique des planètes in 1784, and in the third volume of the Mécanique céleste . In this work, Laplace completely determined

6943-418: The electrostatic potential associated to a charge distribution q , then the charge distribution itself is given by the negative of the Laplacian of φ : q = − ε 0 Δ φ , {\displaystyle q=-\varepsilon _{0}\Delta \varphi ,} where ε 0 is the electric constant . This is a consequence of Gauss's law . Indeed, if V

7074-684: The gradient ( ∇ f {\displaystyle \nabla f} ). Thus if f {\displaystyle f} is a twice-differentiable real-valued function , then the Laplacian of f {\displaystyle f} is the real-valued function defined by: where the latter notations derive from formally writing: ∇ = ( ∂ ∂ x 1 , … , ∂ ∂ x n ) . {\displaystyle \nabla =\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right).} Explicitly,

7205-408: The latitude φ and longitude λ directions, respectively) satisfy Laplace's tidal equations : where Ω is the angular frequency of the planet's rotation, g is the planet's gravitational acceleration at the mean ocean surface, a is the planetary radius, and U is the external gravitational tidal-forcing potential . William Thomson (Lord Kelvin) rewrote Laplace's momentum terms using

7336-620: The origin of the Solar System and was one of the first scientists to suggest an idea similar to that of a black hole , with Stephen Hawking stating that "Laplace essentially predicted the existence of black holes". Laplace is regarded as one of the greatest scientists of all time. Sometimes referred to as the French Newton or Newton of France , he has been described as possessing a phenomenal natural mathematical faculty superior to that of almost all of his contemporaries. He

7467-565: The specific heat of various bodies, and the expansion of metals with increasing temperature. They also measured the boiling points of ethanol and ether under pressure. Laplace further impressed the Marquis de Condorcet , and already by 1771 Laplace felt entitled to membership in the French Academy of Sciences . However, that year admission went to Alexandre-Théophile Vandermonde and in 1772 to Jacques Antoine Joseph Cousin. Laplace

7598-481: The stability of the Solar System ; today the Solar System is understood to be generally chaotic at fine scales, although currently fairly stable on coarse scale. One particular problem from observational astronomy was the apparent instability whereby Jupiter's orbit appeared to be shrinking while that of Saturn was expanding. The problem had been tackled by Leonhard Euler in 1748, and Joseph Louis Lagrange in 1763, but without success. In 1776, Laplace published

7729-497: The velocity potential of a fluid had been obtained some years previously by Leonhard Euler . Laplace's subsequent work on gravitational attraction was based on this result. The quantity ∇ V has been termed the concentration of V and its value at any point indicates the "excess" of the value of V there over its mean value in the neighbourhood of the point. Laplace's equation , a special case of Poisson's equation , appears ubiquitously in mathematical physics. The concept of

7860-522: The Comte de Colbert-Laplace. Others had been destroyed earlier, when his house at Arcueil near Paris was looted in 1871. Laplace was born in Beaumont-en-Auge , Normandy on 23 March 1749, a village four miles west of Pont l'Évêque . According to W. W. Rouse Ball , his father, Pierre de Laplace, owned and farmed the small estates of Maarquis. His great-uncle, Maitre Oliver de Laplace, had held

7991-514: The Jupiter–Saturn system because of the near approach to commensurability of the mean motions of Jupiter and Saturn. In this context commensurability means that the ratio of the two planets' mean motions is very nearly equal to a ratio between a pair of small whole numbers. Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's. The corresponding difference between multiples of

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8122-959: The Laplace operator can be defined as: ∇ 2 f ( x → ) = lim R → 0 2 n R 2 ( f s h e l l R − f ( x → ) ) = lim R → 0 2 n A n − 1 R 2 + n ∫ s h e l l R f ( r → ) − f ( x → ) d r n − 1 {\displaystyle \nabla ^{2}f({\overrightarrow {x}})=\lim _{R\rightarrow 0}{\frac {2n}{R^{2}}}(f_{shell_{R}}-f({\overrightarrow {x}}))=\lim _{R\rightarrow 0}{\frac {2n}{A_{n-1}R^{2+n}}}\int _{shell_{R}}f({\overrightarrow {r}})-f({\overrightarrow {x}})dr^{n-1}} Where n {\displaystyle n}

8253-473: The Laplacian of f is thus the sum of all the unmixed second partial derivatives in the Cartesian coordinates x i : As a second-order differential operator, the Laplace operator maps C functions to C functions for k ≥ 2 . It is a linear operator Δ : C ( R ) → C ( R ) , or more generally, an operator Δ : C (Ω) → C (Ω) for any open set Ω ⊆ R . Alternatively,

8384-446: The Laplacian operator has been used for various tasks, such as blob and edge detection . The Laplacian is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology . The Laplace operator is a second-order differential operator in the n -dimensional Euclidean space , defined as the divergence ( ∇ ⋅ {\displaystyle \nabla \cdot } ) of

8515-533: The Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with D i {\displaystyle D_{i}} as the partial derivative symbol with respect to the i -th variable. For instance, one would write D 1 f ( 17 , u + v , v 2 ) {\displaystyle D_{1}f(17,u+v,v^{2})} for

8646-406: The Solar System is given in his Mécanique céleste published in five volumes. The first two volumes, published in 1799, contain methods for calculating the motions of the planets, determining their figures, and resolving tidal problems. The third and fourth volumes, published in 1802 and 1805, contain applications of these methods, and several astronomical tables. The fifth volume, published in 1825,

8777-672: The age of sixteen Laplace left the "School of the Duke of Orleans" in Beaumont and went to the University of Caen , where he appears to have studied for five years and was a member of the Sphinx. The École Militaire of Beaumont did not replace the old school until 1776. His parents, Pierre Laplace and Marie-Anne Sochon, were from comfortable families. The Laplace family was involved in agriculture until at least 1750, but Pierre Laplace senior

8908-472: The aid of Laplace's discoveries, the tables of the motions of Jupiter and Saturn could at last be made much more accurate. It was on the basis of Laplace's theory that Delambre computed his astronomical tables. Laplace now set himself the task to write a work which should "offer a complete solution of the great mechanical problem presented by the Solar System, and bring theory to coincide so closely with observation that empirical equations should no longer find

9039-443: The attraction of a spheroid on a particle outside it. This is memorable for the introduction into analysis of spherical harmonics or Laplace's coefficients , and also for the development of the use of what we would now call the gravitational potential in celestial mechanics . In 1783, in a paper sent to the Académie , Adrien-Marie Legendre had introduced what are now known as associated Legendre functions . If two points in

9170-404: The attraction of a spheroid on a particle outside it. This is known for the introduction into analysis of the potential, a useful mathematical concept of broad applicability to the physical sciences. Laplace was a supporter of the corpuscle theory of light of Newton. In the fourth edition of Mécanique Céleste , Laplace assumed that short-ranged molecular forces were responsible for refraction of

9301-622: The average value of f {\displaystyle f} over the ball with radius h {\displaystyle h} centered at p {\displaystyle p} is: f ¯ B ( p , h ) = f ( p ) + Δ f ( p ) 2 ( n + 2 ) h 2 + o ( h 2 ) for h → 0 {\displaystyle {\overline {f}}_{B}(p,h)=f(p)+{\frac {\Delta f(p)}{2(n+2)}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0} Similarly,

9432-633: The average value of f {\displaystyle f} over the sphere (the boundary of a ball) with radius h {\displaystyle h} centered at p {\displaystyle p} is: f ¯ S ( p , h ) = f ( p ) + Δ f ( p ) 2 n h 2 + o ( h 2 ) for h → 0. {\displaystyle {\overline {f}}_{S}(p,h)=f(p)+{\frac {\Delta f(p)}{2n}}h^{2}+o(h^{2})\quad {\text{for}}\;\;h\to 0.} If φ denotes

9563-610: The boundary of V . By the divergence theorem , ∫ V div ⁡ ∇ u d V = ∫ S ∇ u ⋅ n d S = 0. {\displaystyle \int _{V}\operatorname {div} \nabla u\,dV=\int _{S}\nabla u\cdot \mathbf {n} \,dS=0.} Since this holds for all smooth regions V , one can show that it implies: div ⁡ ∇ u = Δ u = 0. {\displaystyle \operatorname {div} \nabla u=\Delta u=0.} The left-hand side of this equation

9694-504: The boundary of U . Then: d d ε | ε = 0 E ( f + ε u ) = ∫ U ∇ f ⋅ ∇ u d x = − ∫ U u Δ f d x {\displaystyle \left.{\frac {d}{d\varepsilon }}\right|_{\varepsilon =0}E(f+\varepsilon u)=\int _{U}\nabla f\cdot \nabla u\,dx=-\int _{U}u\,\Delta f\,dx} where

9825-608: The components of the vector field A {\displaystyle \mathbf {A} } , and ∇ 2 {\displaystyle \nabla ^{2}} just on the left of each vector field component is the (scalar) Laplace operator. This can be seen to be a special case of Lagrange's formula; see Vector triple product . For expressions of the vector Laplacian in other coordinate systems see Del in cylindrical and spherical coordinates . The Laplacian of any tensor field T {\displaystyle \mathbf {T} } ("tensor" includes scalar and vector)

9956-427: The corpuscles of light. Laplace and Étienne-Louis Malus also showed that Huygens principle of double refraction could be recovered from the principle of least action on light particles. However in 1815, Augustin-Jean Fresnel presented a new a wave theory for diffraction to a commission of the French Academy with the help of François Arago . Laplace was one of the commission members and they ultimately awarded

10087-482: The details. The work was carried forward in a more finely tuned form in Félix Tisserand 's Traité de mécanique céleste (1889–1896), but Laplace's treatise remains a standard authority. In the years 1784–1787, Laplace produced some memoirs of exceptional power. The significant among these was one issued in 1784, and reprinted in the third volume of the Mécanique céleste . In this work he completely determined

10218-710: The direction of derivative is not repeated, it is called a mixed partial derivative . If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem : ∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i . {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.} For

10349-711: The example described above, while the expression D 1 f {\displaystyle D_{1}f} represents the partial derivative function with respect to the first variable. For higher order partial derivatives, the partial derivative (function) of D i f {\displaystyle D_{i}f} with respect to the j -th variable is denoted D j ( D i f ) = D i , j f {\displaystyle D_{j}(D_{i}f)=D_{i,j}f} . That is, D j ∘ D i = D i , j {\displaystyle D_{j}\circ D_{i}=D_{i,j}} , so that

10480-411: The fact that every function of the points on a circle can be expanded as a series of them. Laplace, with scant regard for credit to Legendre, made the non-trivial extension of the result to three dimensions to yield a more general set of functions, the spherical harmonics or Laplace coefficients . The latter term is not in common use now. This paper is also remarkable for the development of

10611-479: The familiar form. If T {\displaystyle \mathbf {T} } is a vector (a tensor of first degree), the gradient is a covariant derivative which results in a tensor of second degree, and the divergence of this is again a vector. The formula for the vector Laplacian above may be used to avoid tensor math and may be shown to be equivalent to the divergence of the Jacobian matrix shown below for

10742-1104: The first and second term, these expressions read Δ f = ∂ 2 f ∂ r 2 + 2 r ∂ f ∂ r + 1 r 2 sin ⁡ θ ( cos ⁡ θ ∂ f ∂ θ + sin ⁡ θ ∂ 2 f ∂ θ 2 ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}\sin \theta }}\left(\cos \theta {\frac {\partial f}{\partial \theta }}+\sin \theta {\frac {\partial ^{2}f}{\partial \theta ^{2}}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} where φ represents

10873-831: The first equality is due to the divergence theorem . Since the electrostatic field is the (negative) gradient of the potential, this gives: − ∫ V div ⁡ ( grad ⁡ φ ) d V = 1 ε 0 ∫ V q d V . {\displaystyle -\int _{V}\operatorname {div} (\operatorname {grad} \varphi )\,dV={\frac {1}{\varepsilon _{0}}}\int _{V}q\,dV.} Since this holds for all regions V , we must have div ⁡ ( grad ⁡ φ ) = − 1 ε 0 q {\displaystyle \operatorname {div} (\operatorname {grad} \varphi )=-{\frac {1}{\varepsilon _{0}}}q} The same approach implies that

11004-2203: The following examples, let f be a function in x , y , and z . First-order partial derivatives: ∂ f ∂ x = f x ′ = ∂ x f . {\displaystyle {\frac {\partial f}{\partial x}}=f'_{x}=\partial _{x}f.} Second-order partial derivatives: ∂ 2 f ∂ x 2 = f x x ″ = ∂ x x f = ∂ x 2 f . {\displaystyle {\frac {\partial ^{2}f}{\partial x^{2}}}=f''_{xx}=\partial _{xx}f=\partial _{x}^{2}f.} Second-order mixed derivatives : ∂ 2 f ∂ y ∂ x = ∂ ∂ y ( ∂ f ∂ x ) = ( f x ′ ) y ′ = f x y ″ = ∂ y x f = ∂ y ∂ x f . {\displaystyle {\frac {\partial ^{2}f}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial f}{\partial x}}\right)=(f'_{x})'_{y}=f''_{xy}=\partial _{yx}f=\partial _{y}\partial _{x}f.} Higher-order partial and mixed derivatives: ∂ i + j + k f ∂ x i ∂ y j ∂ z k = f ( i , j , k ) = ∂ x i ∂ y j ∂ z k f . {\displaystyle {\frac {\partial ^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z^{k}}}=f^{(i,j,k)}=\partial _{x}^{i}\partial _{y}^{j}\partial _{z}^{k}f.} When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics ,

11135-566: The function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like ∂ f ( x , y , z ) ∂ x {\displaystyle {\frac {\partial f(x,y,z)}{\partial x}}} is used for the function, while ∂ f ( u , v , w ) ∂ u {\displaystyle {\frac {\partial f(u,v,w)}{\partial u}}} might be used for

11266-878: The gradient of a vector: ∇ T = ( ∇ T x , ∇ T y , ∇ T z ) = [ T x x T x y T x z T y x T y y T y z T z x T z y T z z ] ,  where  T u v ≡ ∂ T u ∂ v . {\displaystyle \nabla \mathbf {T} =(\nabla T_{x},\nabla T_{y},\nabla T_{z})={\begin{bmatrix}T_{xx}&T_{xy}&T_{xz}\\T_{yx}&T_{yy}&T_{yz}\\T_{zx}&T_{zy}&T_{zz}\end{bmatrix}},{\text{ where }}T_{uv}\equiv {\frac {\partial T_{u}}{\partial v}}.} And, in

11397-436: The higher-order terms, up to and including the cubic . Using this more exact analysis, Laplace concluded that any two planets and the Sun must be in mutual equilibrium and thereby launched his work on the stability of the Solar System. Gerald James Whitrow described the achievement as "the most important advance in physical astronomy since Newton". Laplace had a wide knowledge of all sciences and dominated all discussions in

11528-428: The history of scientific development. Partial derivative In mathematics , a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative , in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry . The partial derivative of

11659-401: The hypothesis, the Solar System evolved from a globular mass of incandescent gas rotating around an axis through its centre of mass . As it cooled, this mass contracted, and successive rings broke off from its outer edge. These rings in their turn cooled, and finally condensed into the planets, while the Sun represented the central core which was still left. On this view, Laplace predicted that

11790-405: The idea of the scalar potential . The gravitational force acting on a body is, in modern language, a vector , having magnitude and direction. A potential function is a scalar function that defines how the vectors will behave. A scalar function is computationally and conceptually easier to deal with than a vector function. Alexis Clairaut had first suggested the idea in 1743 while working on

11921-441: The identification and explanation of the perturbations now known as the "great Jupiter–Saturn inequality". Laplace solved a longstanding problem in the study and prediction of the movements of these planets. He showed by general considerations, first, that the mutual action of two planets could never cause large changes in the eccentricities and inclinations of their orbits; but then, even more importantly, that peculiarities arose in

12052-597: The inverse of the Laplacian (which is compact, by the Poincaré inequality and the Rellich–Kondrachov theorem ). It can also be shown that the eigenfunctions are infinitely differentiable functions. More generally, these results hold for the Laplace–Beltrami operator on any compact Riemannian manifold with boundary, or indeed for the Dirichlet eigenvalue problem of any elliptic operator with smooth coefficients on

12183-498: The large amphidromic systems in the world's ocean basins and explains the oceanic tides that are actually observed. The equilibrium theory, based on the gravitational gradient from the Sun and Moon but ignoring the Earth's rotation, the effects of continents, and other important effects, could not explain the real ocean tides. Since measurements have confirmed the theory, many things have possible explanations now, like how

12314-679: The last equality follows using Green's first identity . This calculation shows that if Δ f = 0 , then E is stationary around f . Conversely, if E is stationary around f , then Δ f = 0 by the fundamental lemma of calculus of variations . The Laplace operator in two dimensions is given by: In Cartesian coordinates , Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}} where x and y are

12445-416: The masterpieces of French literature, though it is not altogether reliable for the later periods of which it treats. Laplace developed the nebular hypothesis of the formation of the Solar System, first suggested by Emanuel Swedenborg and expanded by Immanuel Kant . This hypothesis remains the most widely accepted model in the study of the origin of planetary systems. According to Laplace's description of

12576-495: The mean motions, (2 n J − 5 n S ) , corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period. As a result, the integrated perturbations with this period are disproportionately large, about 0.8° degrees of arc in orbital longitude for Saturn and about 0.3° for Jupiter. Further developments of these theorems on planetary motion were given in his two memoirs of 1788 and 1789, but with

12707-416: The method of separation of variables into a radial part, depending solely on distance from the centre point, and an angular or spherical part. The solution to the spherical part of the equation can be expressed as a series of Laplace's spherical harmonics, simplifying practical computation. Laplace presented a memoir on planetary inequalities in three sections, in 1784, 1785, and 1786. This dealt mainly with

12838-450: The more distant planets would be older than those nearer the Sun. As mentioned, the idea of the nebular hypothesis had been outlined by Immanuel Kant in 1755, who had also suggested "meteoric aggregations" and tidal friction as causes affecting the formation of the Solar System. Laplace was probably aware of this, but, like many writers of his time, he generally did not reference the work of others. Laplace's analytical discussion of

12969-434: The negative of the Laplacian of the gravitational potential is the mass distribution . Often the charge (or mass) distribution are given, and the associated potential is unknown. Finding the potential function subject to suitable boundary conditions is equivalent to solving Poisson's equation . Another motivation for the Laplacian appearing in physics is that solutions to Δ f = 0 in a region U are functions that make

13100-416: The next seventeen years, 1771–1787, he produced much of his original work in astronomy. From 1780 to 1784, Laplace and French chemist Antoine Lavoisier collaborated on several experimental investigations, designing their own equipment for the task. In 1783 they published their joint paper, Memoir on Heat , in which they discussed the kinetic theory of molecular motion. In their experiments they measured

13231-425: The notation, such as in: f x ′ ( x , y , … ) , ∂ f ∂ x ( x , y , … ) . {\displaystyle f'_{x}(x,y,\ldots ),{\frac {\partial f}{\partial x}}(x,y,\ldots ).} The symbol used to denote partial derivatives is ∂ . One of the first known uses of this symbol in mathematics

13362-431: The partial derivative of z {\displaystyle z} with respect to x {\displaystyle x} is denoted as ∂ z ∂ x . {\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by

13493-401: The partial derivative of f with respect to x , holding y and z constant, is often expressed as ( ∂ f ∂ x ) y , z . {\displaystyle \left({\frac {\partial f}{\partial x}}\right)_{y,z}.} Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of

13624-468: The particular question he was discussing. While Newton explained the tides by describing the tide-generating forces and Bernoulli gave a description of the static reaction of the waters on Earth to the tidal potential, the dynamic theory of tides , developed by Laplace in 1775, describes the ocean's real reaction to tidal forces . Laplace's theory of ocean tides took into account friction , resonance and natural periods of ocean basins. It predicted

13755-440: The possibility of a mathematical solution to the whole, even concluding that periodic divine intervention was necessary to guarantee the stability of the Solar System. Dispensing with the hypothesis of divine intervention would be a major activity of Laplace's scientific life. It is now generally regarded that Laplace's methods on their own, though vital to the development of the theory, are not sufficiently precise to demonstrate

13886-524: The possible gravitational potentials in regions of vacuum . The Laplacian occurs in many differential equations describing physical phenomena. Poisson's equation describes electric and gravitational potentials ; the diffusion equation describes heat and fluid flow ; the wave equation describes wave propagation ; and the Schrödinger equation describes the wave function in quantum mechanics . In image processing and computer vision ,

14017-461: The press, says that Laplace himself was frequently unable to recover the details in the chain of reasoning, and, if satisfied that the conclusions were correct, he was content to insert the phrase, " Il est aisé à voir que... " ("It is easy to see that..."). The Mécanique céleste is not only the translation of Newton's Principia Mathematica into the language of differential calculus , but it completes parts of which Newton had been unable to fill in

14148-1407: The radial distance and θ the angle. In three dimensions, it is common to work with the Laplacian in a variety of different coordinate systems. In Cartesian coordinates , Δ f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\displaystyle \Delta f={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.} In cylindrical coordinates , Δ f = 1 ρ ∂ ∂ ρ ( ρ ∂ f ∂ ρ ) + 1 ρ 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2 , {\displaystyle \Delta f={\frac {1}{\rho }}{\frac {\partial }{\partial \rho }}\left(\rho {\frac {\partial f}{\partial \rho }}\right)+{\frac {1}{\rho ^{2}}}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}},} where ρ {\displaystyle \rho } represents

14279-1934: The radial distance, φ the azimuth angle and z the height. In spherical coordinates : Δ f = 1 r 2 ∂ ∂ r ( r 2 ∂ f ∂ r ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {1}{r^{2}}}{\frac {\partial }{\partial r}}\left(r^{2}{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} or Δ f = 1 r ∂ 2 ∂ r 2 ( r f ) + 1 r 2 sin ⁡ θ ∂ ∂ θ ( sin ⁡ θ ∂ f ∂ θ ) + 1 r 2 sin 2 ⁡ θ ∂ 2 f ∂ φ 2 , {\displaystyle \Delta f={\frac {1}{r}}{\frac {\partial ^{2}}{\partial r^{2}}}(rf)+{\frac {1}{r^{2}\sin \theta }}{\frac {\partial }{\partial \theta }}\left(\sin \theta {\frac {\partial f}{\partial \theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\frac {\partial ^{2}f}{\partial \varphi ^{2}}},} by expanding

14410-856: The repeated indices is implied , g is the inverse metric tensor and Γ mn are the Christoffel symbols for the selected coordinates. In arbitrary curvilinear coordinates in N dimensions ( ξ , ..., ξ ), we can write the Laplacian in terms of the inverse metric tensor , g i j {\displaystyle g^{ij}} : Δ = 1 det g ∂ ∂ ξ i ( det g g i j ∂ ∂ ξ j ) , {\displaystyle \Delta ={\frac {1}{\sqrt {\det g}}}{\frac {\partial }{\partial \xi ^{i}}}\left({\sqrt {\det g}}g^{ij}{\frac {\partial }{\partial \xi ^{j}}}\right),} from

14541-411: The right. Below, we see how the function looks on the plane y = 1 . By finding the derivative of the equation while assuming that y is a constant, we find that the slope of f at the point ( x , y ) is: ∂ z ∂ x = 2 x + y . {\displaystyle {\frac {\partial z}{\partial x}}=2x+y.} So at (1, 1) , by substitution,

14672-838: The same manner, a dot product , which evaluates to a vector, of a vector by the gradient of another vector (a tensor of 2nd degree) can be seen as a product of matrices: A ⋅ ∇ B = [ A x A y A z ] ∇ B = [ A ⋅ ∇ B x A ⋅ ∇ B y A ⋅ ∇ B z ] . {\displaystyle \mathbf {A} \cdot \nabla \mathbf {B} ={\begin{bmatrix}A_{x}&A_{y}&A_{z}\end{bmatrix}}\nabla \mathbf {B} ={\begin{bmatrix}\mathbf {A} \cdot \nabla B_{x}&\mathbf {A} \cdot \nabla B_{y}&\mathbf {A} \cdot \nabla B_{z}\end{bmatrix}}.} This identity

14803-476: The stability of the Solar System . The two disciplines would always be interlinked in his mind. "Laplace took probability as an instrument for repairing defects in knowledge." Laplace's work on probability and statistics is discussed below with his mature work on the analytic theory of probabilities. Sir Isaac Newton had published his Philosophiæ Naturalis Principia Mathematica in 1687 in which he gave

14934-1110: The standard Cartesian coordinates of the xy -plane. In polar coordinates , Δ f = 1 r ∂ ∂ r ( r ∂ f ∂ r ) + 1 r 2 ∂ 2 f ∂ θ 2 = ∂ 2 f ∂ r 2 + 1 r ∂ f ∂ r + 1 r 2 ∂ 2 f ∂ θ 2 , {\displaystyle {\begin{aligned}\Delta f&={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}\\&={\frac {\partial ^{2}f}{\partial r^{2}}}+{\frac {1}{r}}{\frac {\partial f}{\partial r}}+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}},\end{aligned}}} where r represents

15065-469: The subject. Here Laplace's brilliance as a mathematician was quickly recognised and while still at Caen he wrote a memoir Sur le Calcul integral aux differences infiniment petites et aux differences finies . This provided the first correspondence between Laplace and Lagrange. Lagrange was the senior by thirteen years, and had recently founded in his native city Turin a journal named Miscellanea Taurinensia , in which many of his early works were printed and it

15196-537: The term "potential function" was not actually used (to refer to a function V of the coordinates of space in Laplace's sense) until George Green 's 1828 An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism . Laplace applied the language of calculus to the potential function and showed that it always satisfies the differential equation : An analogous result for

15327-1164: The term with the vector Laplacian of the velocity field μ ( ∇ 2 v ) {\displaystyle \mu \left(\nabla ^{2}\mathbf {v} \right)} represents the viscous stresses in the fluid. Another example is the wave equation for the electric field that can be derived from Maxwell's equations in the absence of charges and currents: ∇ 2 E − μ 0 ϵ 0 ∂ 2 E ∂ t 2 = 0. {\displaystyle \nabla ^{2}\mathbf {E} -\mu _{0}\epsilon _{0}{\frac {\partial ^{2}\mathbf {E} }{\partial t^{2}}}=0.} This equation can also be written as: ◻ E = 0 , {\displaystyle \Box \,\mathbf {E} =0,} where ◻ ≡ 1 c 2 ∂ 2 ∂ t 2 − ∇ 2 , {\displaystyle \Box \equiv {\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\nabla ^{2},}

15458-422: The tides interact with deep sea ridges and chains of seamounts give rise to deep eddies that transport nutrients from the deep to the surface. The equilibrium tide theory calculates the height of the tide wave of less than half a meter, while the dynamic theory explains why tides are up to 15 meters. Satellite observations confirm the accuracy of the dynamic theory, and the tides worldwide are now measured to within

15589-477: The title of Chirurgien Royal. It would seem that from a pupil he became an usher in the school at Beaumont; but, having procured a letter of introduction to d'Alembert , he went to Paris to advance his fortune. However, Karl Pearson is scathing about the inaccuracies in Rouse Ball's account and states: Indeed Caen was probably in Laplace's day the most intellectually active of all the towns of Normandy. It

15720-442: The value of the function at the point ( x , y , z ) = ( u , v , w ) {\displaystyle (x,y,z)=(u,v,w)} . However, this convention breaks down when we want to evaluate the partial derivative at a point like ( x , y , z ) = ( 17 , u + v , v 2 ) {\displaystyle (x,y,z)=(17,u+v,v^{2})} . In such

15851-417: The variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that D i , j = D j , i {\displaystyle D_{i,j}=D_{j,i}} as long as comparatively mild regularity conditions on f are satisfied. An important example of

15982-641: Was Napoleon's examiner when Napoleon graduated from the École Militaire in Paris in 1785. Laplace became a count of the Empire in 1806 and was named a marquis in 1817, after the Bourbon Restoration . Some details of Laplace's life are not known, as records of it were burned in 1925 with the family château in Saint Julien de Mailloc , near Lisieux , the home of his great-great-grandson

16113-438: Was a French scholar whose work was important to the development of engineering , mathematics , statistics , physics , astronomy , and philosophy . He summarized and extended the work of his predecessors in his five-volume Mécanique céleste ( Celestial Mechanics ) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus , opening up a broader range of problems. In statistics,

16244-492: Was already starting to think about the mathematical and philosophical concepts of probability and statistics. However, before his election to the Académie in 1773, he had already drafted two papers that would establish his reputation. The first, Mémoire sur la probabilité des causes par les événements was ultimately published in 1774 while the second paper, published in 1776, further elaborated his statistical thinking and also began his systematic work on celestial mechanics and

16375-553: Was also a cider merchant and syndic of the town of Beaumont. Pierre Simon Laplace attended a school in the village run at a Benedictine priory , his father intending that he be ordained in the Roman Catholic Church . At sixteen, to further his father's intention, he was sent to the University of Caen to read theology. At the university, he was mentored by two enthusiastic teachers of mathematics, Christophe Gadbled and Pierre Le Canu, who awoke his zeal for

16506-565: Was disgruntled, and early in 1773 d'Alembert wrote to Lagrange in Berlin to ask if a position could be found for Laplace there. However, Condorcet became permanent secretary of the Académie in February and Laplace was elected associate member on 31 March, at age 24. In 1773 Laplace read his paper on the invariability of planetary motion in front of the Academy des Sciences. That March he

16637-566: Was elected to the academy, a place where he conducted the majority of his science. On 15 March 1788, at the age of thirty-nine, Laplace married Marie-Charlotte de Courty de Romanges, an eighteen-year-old girl from a "good" family in Besançon . The wedding was celebrated at Saint-Sulpice, Paris . The couple had a son, Charles-Émile (1789–1874), and a daughter, Sophie-Suzanne (1792–1813). Laplace's early published work in 1771 started with differential equations and finite differences but he

16768-580: Was here that Laplace was educated and was provisionally a professor. It was here he wrote his first paper published in the Mélanges of the Royal Society of Turin, Tome iv. 1766–1769, at least two years before he went at 22 or 23 to Paris in 1771. Thus before he was 20 he was in touch with Lagrange in Turin . He did not go to Paris a raw self-taught country lad with only a peasant background! In 1765 at

16899-419: Was in the fourth volume of this series that Laplace's paper appeared. About this time, recognising that he had no vocation for the priesthood, he resolved to become a professional mathematician. Some sources state that he then broke with the church and became an atheist. Laplace did not graduate in theology but left for Paris with a letter of introduction from Le Canu to Jean le Rond d'Alembert who at that time

17030-442: Was supreme in scientific circles. According to his great-great-grandson, d'Alembert received him rather poorly, and to get rid of him gave him a thick mathematics book, saying to come back when he had read it. When Laplace came back a few days later, d'Alembert was even less friendly and did not hide his opinion that it was impossible that Laplace could have read and understood the book. But upon questioning him, he realised that it

17161-431: Was true, and from that time he took Laplace under his care. Another account is that Laplace solved overnight a problem that d'Alembert set him for submission the following week, then solved a harder problem the following night. D'Alembert was impressed and recommended him for a teaching place in the École Militaire . With a secure income and undemanding teaching, Laplace now threw himself into original research and for

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