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D: divergence, C: curl, G: gradient, L: Laplacian, CC: curl of curl. Each arrow is labeled with the result of an identity, specifically, the result of applying the operator at the arrow's tail to the operator at its head. The blue circle in the middle means curl of curl exists, whereas the other two red circles (dashed) mean that DD and GG do ...
In vector calculus, the curl, also known as rotor, is a vector operator that describes the infinitesimal circulation of a vector field in three-dimensional Euclidean space. The curl at a point in the field is represented by a vector whose length and direction denote the magnitude and axis of the maximum circulation. [1]
The divergence of the curl of any vector field (in three dimensions) is equal to zero: ∇ ⋅ ( ∇ × F ) = 0. {\displaystyle \nabla \cdot (\nabla \times \mathbf {F} )=0.} If a vector field F with zero divergence is defined on a ball in R 3 , then there exists some vector field G on the ball with F = curl G .
Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc. This page was last edited on 12 October 2024, at 11:14 ...
Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators.
Vector calculus or vector analysis is a branch of mathematics concerned with the differentiation and integration of vector fields, primarily in three-dimensional Euclidean space, . [1] The term vector calculus is sometimes used as a synonym for the broader subject of multivariable calculus, which spans vector calculus as well as partial differentiation and multiple integration.
As we have only specified the curl of A, we are free to define the divergence, and choose the following: = which is called the Lorenz gauge condition. The previous expression for A now reduces to ∇ 2 A + k 2 A = − J {\displaystyle \nabla ^{2}\mathbf {A} +k^{2}\mathbf {A} =-\mathbf {J} \,} which is the vector Helmholtz equation .
The term incompressible is used because a non-zero divergence corresponds to the presence of sources and sinks in analogy with a fluid. The concepts of conservative and incompressible vector fields generalize to n dimensions, because gradient and divergence generalize to n dimensions; curl is defined only in three dimensions, thus the concept ...