Misplaced Pages

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75-502: 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

150-506: 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}} }

225-402: 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 (

300-419: 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

375-533: 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 ,

450-392: 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})}

525-2472: 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

600-482: 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

675-427: 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

750-671: 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

825-407: 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

SECTION 10

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900-505: 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 +

975-563: 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

1050-506: 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 (

1125-479: 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 (

1200-745: 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

1275-437: 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}} ,

1350-886: 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 Too Many Requests If you report this error to

1425-603: 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

1500-602: Is ∂ . One of the first known uses of this symbol in mathematics 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} }

1575-727: 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

SECTION 20

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1650-562: 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

1725-455: 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

1800-469: 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

1875-598: 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

1950-471: 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

2025-657: 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 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

2100-607: 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

2175-570: 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

2250-518: 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

2325-402: 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

2400-422: 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

2475-434: 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

2550-470: 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}}

2625-407: 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

2700-407: 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 )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x}

2775-460: 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

2850-643: 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

2925-763: 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

3000-1125: 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

3075-418: 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

Laplace operator - Misplaced Pages Continue

3150-684: 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,

3225-959: 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}

3300-468: 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,

3375-446: 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

3450-524: 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

3525-486: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.132 via cp1112 cp1112, Varnish XID 389064917 Upstream caches: cp1112 int Error: 429, Too Many Requests at Fri, 29 Nov 2024 05:34:15 GMT Partial derivative In mathematics , a partial derivative of a function of several variables is its derivative with respect to one of those variables, with

3600-622: 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,

3675-633: 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

3750-610: 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

3825-504: 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

SECTION 50

#1732858455700

3900-608: 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)

3975-710: 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

4050-711: 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

4125-474: 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

4200-1104: 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

4275-831: 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

4350-2203: 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 ,

4425-566: 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

4500-878: 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

4575-587: 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

SECTION 60

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4650-679: 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

4725-434: 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

4800-394: 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 a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x}

4875-401: 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

4950-518: 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 ,

5025-1407: 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

5100-1934: 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

5175-850: 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

5250-411: 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,

5325-838: 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

5400-1110: 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

5475-1164: 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},}

5550-442: 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

5625-417: 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

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