In vector calculus, an irrotational or conservative vector field is a vector field whose curl is zero. If the field is denoted as v, then Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. ...

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There is an identity of vector calculus which states that the curl of any gradient is zero: Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

where φ is a scalar field.

Conversely, any irrotational field can be expressed as the gradient of a scalar potential: It has been suggested that this article or section be merged with Potential. ...

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If, in addition to being irrotational, a field is also incompressible, then the field is called a Laplacian field. In fluid mechanics, an incompressible fluid is a fluid whose density (often represented by the Greek letter ρ) is constant: it is the same throughout the field and it does not change through time. ... In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. ...

In fluid mechanics, an irrotational field is practically synonymous with a lamellar field. The adjective "irrotational" implies that irrotational fluid flow (whose velocity field is irrotational) has no rotational component: the fluid does not move in circular or helical motions; it does not form vortices. Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases. ... In vector analysis and in fluid dynamics, a lamellar vector field is a vector field with no rotational component. ... Vortex created by the passage of an aircraft wing, revealed by coloured smoke A vortex is a spinning turbulent flow (or any spiral whirling motion) with closed streamlines. ...

From the zero curl definition of an irrotational field, it can be deduced, by means of Stokes' theorem, that the circulation of any closed loop in the field is zero: The Stokes theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. ... The word circulation can mean the following: The transport of blood through the circulatory system. ...

where A is the area enclosed by loop S. This lack of circulation means that irrotational field lines (streamlines of irrotational flow) do not form loops (or helices). In fluid dynamics, a streamline is a line which is everywhere tangent to the velocity of the flow. ... A helix (pl: helices), from the Greek word έλικας/έλιξ, is a twisted shape like a spring, screw or a spiral staircase. ...