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Solenoidal vector field

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	Topics Solenoidal vector field Topic Home In vector calculusVector calculus Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions.... a solenoidal vector field (also known as an incompressible vector field) is a vector fieldVector field In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean s... v with divergenceDivergence In vector calculus, the divergence is an operator that measures a vector field's tendency to originate from or converge upo... zero: The fundamental theorem of vector calculusHelmholtz decomposition In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calc... states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field v has only a vector potentialVector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field.... component, because the definition of the vector potential A as: automatically results in the identityVector calculus identities The following identities are important in vector calculus: ... (as can be shown, for example, using Cartesian coordinates): The converse also holds: for any solenoidal v there exists a vector potential A such that (Strictly speaking, this holds only subject to certain technical conditions on v, see Helmholtz decompositionHelmholtz decomposition In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calc... .) The divergence theoremDivergence theorem In vector calculus, the divergence theorem, also known as Gauss' theorem, Ostrogradsky's theorem, or Ostrogra... , gives the equivalent integral definition of a solenoidal field; namely that for any closed surface , the net total flux through the surface must be zero: , where is the outward normal to each surface element. Etymology Solenoidal has its origin in the Greek word for solenoidSolenoid A solenoid is a loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electrical curre... , which is s?????e?d?? (solenoeides) and meaning pipe-shaped. This contains s???? (solen) or pipe. In the present context of solenoidal it means constrained like in a pipe, so with a fixed volume. Examples the magnetic fieldMagnetic field In physics, a magnetic field is that part of the electromagnetic field that exists when there is a changing electric field.... B is solenoidal (see Maxwell's equationsMaxwell's equations Summary In electromagnetics, Maxwell's equations are a set of four equations, developed by James Clerk Maxwell, that describe the be... ); the velocityVelocity The velocity of an object is simply its speed in a particular direction.... field of an incompressible fluid flow is solenoidal; the electric fieldElectric field In physics, the properties of space that surrounds an electric charge can be described using an electric field or E-field... in regions where ?_e = 0; the current densityCurrent density Current density is a measure of the density of electrical current.... , J, if .