In vector calculus, a Laplacian vector field is a vector field which is both irrotational and incompressible. If the field is denoted as v, then it is described by the following differential equations: 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 fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ... 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 mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

Since the curl of v is zero, it follows that v can be expressed as the gradient of a scalar potential (see irrotational field) φ : 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... A scalar potential is, mathematically, a scalar field whose negative gradient is a given vector field. ... In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ...

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Then, since the divergence of v is also zero, it follows from equation (1) that In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ...

which is equivalent to

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Therefore, the potential of a Laplacian field satisfies Laplace's equation. Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...

See also: potential flow, harmonic function fluid dynamics, potential flow, also known as irrotational flow (of incompressible fluids) is steady flow defined by the equations (zero rotation = no viscosity) (zero divergence = volume conservation) Equivalently, where: v is the vector fluid velocity Φ is the fluid flow potential, scalar × is curl · is divergence. ... In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R (where U is an open subset of Rn) which satisfies Laplaces equation, i. ...