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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, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
The divergence theorem can be used to calculate a flux through a closed surface that fully encloses a volume, ... is the unit circle, , that can be ...
By the divergence theorem, = = 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 is the Laplace operator, and the entire equation Δ u = 0 is known as Laplace's equation .
The gradient of a function is obtained by raising the index of the differential , whose components are given by: =; =; =, = = The divergence of a vector field with components is
[citation needed] This is why the magnetic field, characterized by zero divergence, can be expressed as the curl of a magnetic vector potential. If W is a vector field with curl( W ) = V , then adding any gradient vector field grad( f ) to W will result in another vector field W + grad( f ) such that curl( W + grad( f )) = V as well.
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.
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.