In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. If the field is denoted as v, then Fluid mechanics or fluid dynamics is the study of the macroscopic physical behaviour of fluids . ... 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 Euclidean space. ... cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, Gopher, Telnet, DICT, FILE and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password authentication, file transfer resume, http proxy tunneling and...

.

Since 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 vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change. ...

where φ is a scalar field, it follows that any irrotational field can be expressed as the gradient of a scalar potential: A scalar potential is, mathematically, a scalar field whose negative gradient is a given vector field. ...

.

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. ...

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. 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). You really need to make it way more easier to understand ... This article is about the shape. ...