vector Laplacian

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• Vector Laplacian — In mathematics and physics, the vector Laplace operator, denoted by scriptstyle abla^2, named after Pierre Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian. Whereas the …   Wikipedia

• Vector Laplacian/Proofs — The following is the proof that: abla ^2 left( {mathbf{u ight) = abla left( { abla cdot{mathbf{u} ight) abla imes left( { abla imes {mathbf{u} ight) = leftlangle { abla ^2 u x , abla ^2 u y , abla ^2 u z } ight angle. This is a proof in Cartesian …   Wikipedia

• Vector calculus — Topics in Calculus Fundamental theorem Limits of functions Continuity Mean value theorem Differential calculus  Derivative Change of variables Implicit differentiation Taylor s theorem Related rates …   Wikipedia

• Laplacian operators in differential geometry — In differential geometry there are a number of second order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them. Connection Laplacian The connection Laplacian is a differential… …   Wikipedia

• Laplacian vector field — In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations:: abla imes mathbf{v} = 0, : abla cdot… …   Wikipedia

• Vector operator — A vector operator is a type of differential operator used in vector calculus. Vector operators are defined in terms of del, and include the gradient, divergence, and curl:: operatorname{grad} equiv abla : operatorname{div} equiv abla cdot :… …   Wikipedia

• Vector calculus identities — The following identities are important in vector calculus:ingle operators (summary)This section explicitly lists what some symbols mean for clarity.DivergenceDivergence of a vector fieldFor a vector field mathbf{v} , divergence is generally… …   Wikipedia

• Vector fields in cylindrical and spherical coordinates — Cylindrical coordinate system = Vector fields Vectors are defined in cylindrical coordinates by (ρ,φ,z), where * ρ is the length of the vector projected onto the X Y plane, * φ is the angle of the projected vector with the positive X axis (0 ≤ φ… …   Wikipedia

• Conservative vector field — In vector calculus a conservative vector field is a vector field which is the gradient of a function, known in this context as a scalar potential. Conservative vector fields have the property that the line integral from one point to another is… …   Wikipedia

• Lamellar vector field — In vector analysis and in fluid dynamics, a lamellar vector field is a vector field with no rotational component. That is, if the field is denoted as v, then : abla imes mathbf{v} = 0 .A lamellar field is practically synonymous with an… …   Wikipedia

• Del — For other uses, see Del (disambiguation). ∇ Del operator, represented by the nabla symbol In vector calculus, del is a vector differential operator, usually represented by the nabla symbol . When applied to a function defined on a one dimensional …   Wikipedia