irrotational

In vector calculus a conservative vector field is a vector field which is the gradient of a scalar potential. There are two closely related concepts: path independence and irrotational vector fields. These three properties are equivalent in many 'real-world' applications.

Definition

A vector field is said to be conservative if there exists a scalar field such that



Here denotes the gradient of . When the above equation holds, is called a scalar potential for .

Path independence

A key property of a conservative vector field is that its integral along a path depends only on the endpoints of that path, not the particular route taken. Suppose that is some region of three-dimensional space, and that is a path in with start point and end point . If is a conservative vector field then



This holds as a consequence of the Chain Rule and the Fundamental Theorem of Calculus.

An equivalent formulation of this is to say that



for every closed loop in .

The converse of the above statement is also true. That is, if the circulation of around every closed loop in is zero, then is a conservative vector field.

Irrotational vector fields

A vector field is said to be irrotational if its curl is zero. That is, if



For this reason, such vector fields are sometimes referred to as curl-free vector fields.

It is an identity of vector calculus that for any scalar field :



Therefore every conservative vector field is also an irrotational vector field.

Provided that is a simply-connected region the converse of this is true: every irrotational vector field is also a conservative vector field.

The above statement is not true if is not simply-connected. Let be the usual 3-dimensional space, except with the -axis removed; that is . Now define a vector field by



Then exists and has zero curl at every point in ; that is is irrotational. However the circulation of around the unit circle in the -plane is equal to . Therefore does not have the path independence property discussed above, and is not conservative.

In a simply-connected region an irrotational vector field has the path independence property. This can be proved directly by using Stokes' Theorem.

Irrotational fluids

The flow velocity of a fluid is a vector field, and the vorticity of the fluid is (usually) defined by



If is irrotational then the fluid is said to be an irrotational fluid, or to have irrotational flow. The vorticity of an irrotational fluid is zero.

For a two-dimensional flow the vorticity acts as a measure of the local rotation of fluid elements. Note that the vorticity does not imply anything about the global behaviour of a fluid. It is possible for a fluid traveling in a straight line to have vorticity, and it is possible for a fluid which moves in a circle to be irrotational. For more information see: Vortex.

Conservative forces

If the vector field associated to a force is conservative then the force is said to be a conservative force. The most prominent example of a conservative force is the force of gravity. The gravitational force on a mass due to a mass which is a distance away can be written as



where is the Gravitational Constant and is a unit vector pointing from towards . In this case , where



is the Gravitational potential.

In the case of conservative forces, path independence can be interpreted to mean that the work done in going from a point to a point is independent of the path chosen, and that the work done in going around a closed loop is zero. In other words, the total energy of a particle moving under the influence of conservative forces is conserved.

References

  • George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 6th edition, Elsevier Academic Press (2005)
  • D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press (2005)

See also

Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (some results can only be applied to three dimensions[1]).
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vector field is a construction in vector calculus which associates a vector to every point in a (locally) Euclidean space.

Vector fields are often used in physics to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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A scalar potential is a fundamental concept in vector analysis and physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion with vector potential).
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gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
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A scalar potential is a fundamental concept in vector analysis and physics (the adjective 'scalar' is frequently omitted if there is no danger of confusion with vector potential).
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In calculus, the chain rule is a formula for the derivative of the composite of two functions.

In intuitive terms, if a variable, y, depends on a second variable, u, which in turn depends on a third variable, x, then the rate of change of
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In fluid dynamics, circulation is the line integral around a closed curve of the fluid velocity. Circulation is normally denoted . If is the fluid velocity and is a unit vector along the closed curve :


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cURL is a command line tool for transferring files with URL syntax, supporting FTP, FTPS, HTTP, HTTPS, TFTP, SCP, SFTP, Telnet, DICT, and LDAP. cURL supports HTTPS certificates, HTTP POST, HTTP PUT, FTP uploading, Kerberos, HTTP form based upload, proxies, cookies, user+password
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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In topology, a geometrical object or space is called simply connected (or 1-connected) if it is path-connected and every path between two points can be continuously transformed into every other.
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Stokes' theorem (or Stokes's theorem) in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus.
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In fluid dynamics the flow velocity, or velocity field, of a fluid is a vector field which is used to mathematically describe the motion of the fluid.

Definition

The flow velocity of a fluid is a vector field


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Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "circulation" or "rotation" (or more strictly, the local angular rate of rotation) in a fluid.
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Verification of the Origins of Rotation in Tornadoes Experiment or VORTEX, is a field project that seeks to understand how a tornado is produced by deploying around 18 vehicles that are equipped with customized instruments used to measure and analyze the weather around a
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A conservative force is defined as a force that does not depend on the path taken to increase in potential energy.

Informal definition

Informally, a conservative force can be thought of as a force that conserves mechanical energy.
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Isaac Newton's theory of universal gravitation is a physical law describing the gravitational attraction between massive bodies. It is a part of classical mechanics and was first formulated in Newton's work Philosophiae Naturalis Principia Mathematica, published in 1687.
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gravitational constant, the universal gravitational constant, Newton's constant, and colloquially Big G. The gravitational constant is a physical constant which appears in Newton's law of universal gravitation and in Einstein's theory of general
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A conservative force is defined as a force that does not depend on the path taken to increase in potential energy.

Informal definition

Informally, a conservative force can be thought of as a force that conserves mechanical energy.
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In physics, mechanical work is the amount of energy transferred by a force. Like energy, it is a scalar quantity, with SI units of joules. Heat conduction is not considered to be a form of work, since there is no macroscopically measurable force, only microscopic forces occurring
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In vector calculus a solenoidal vector field is a vector field v with divergence zero:



This condition is satisfied whenever v has a vector potential, because if



then


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In mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field can be resolved into irrotational (curl-free) and solenoidal (divergence-free)
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