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In vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
, a vector potential is a vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a 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 of some force, such as the magnetic field or gravity for...
 whose curl is a given vector field. This is analogous to a scalar potential
Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . Given a vector field F, its scalar potential V is a scalar field whose negative gradient is F,...
, which is a scalar field whose negative gradient
Gradient

In vector calculus, the 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....
 is a given vector field.

Formally, given a vector field v, a vector potential is a vector field A such that

If a vector field v admits a vector potential A, then from the equality (divergence
Divergence

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar....
 of the curl is zero) one obtains which implies that v must be a solenoidal vector field
Solenoidal vector field

In vector calculus a solenoidal vector field is a vector field v with divergence zero:The Helmholtz decomposition states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field....
.

An interesting question is then if any solenoidal vector field admits a vector potential.






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In vector calculus
Vector calculus

Vector calculus is a branch of mathematics concerned with derivative and integral of vector fields. The term "vector calculus" is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial derivative and multiple integral....
, a vector potential is a vector field
Vector field

In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a 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 of some force, such as the magnetic field or gravity for...
 whose curl is a given vector field. This is analogous to a scalar potential
Scalar potential

A scalar potential is a fundamental concept in vector analysis and physics . Given a vector field F, its scalar potential V is a scalar field whose negative gradient is F,...
, which is a scalar field whose negative gradient
Gradient

In vector calculus, the 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....
 is a given vector field.

Formally, given a vector field v, a vector potential is a vector field A such that

If a vector field v admits a vector potential A, then from the equality (divergence
Divergence

In vector calculus, the divergence is an operator that measures the magnitude of a vector field's source or sink at a given point; the divergence of a vector field is a scalar....
 of the curl is zero) one obtains which implies that v must be a solenoidal vector field
Solenoidal vector field

In vector calculus a solenoidal vector field is a vector field v with divergence zero:The Helmholtz decomposition states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field....
.

An interesting question is then if any solenoidal vector field admits a vector potential. The answer is affirmative, if the vector field satisfies certain conditions.

Theorem


Let be solenoidal vector field
Solenoidal vector field

In vector calculus a solenoidal vector field is a vector field v with divergence zero:The Helmholtz decomposition states that any vector field can be expressed as the sum of a conservative vector field and a solenoidal field....
 which is twice continuously differentiable
Smooth function

In mathematical analysis, a differentiability class is a classification of function according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives....
. Assume that v(x) decreases sufficiently fast as ||x||?8. Define

Then, A is a vector potential for v, that is,

A generalization of this theorem is the Helmholtz decomposition
Helmholtz decomposition

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth function, Schwartz space vector field can be resolved into irrotational vector field and solenoidal component vector fields....
 which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field
Irrotational vector field

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....
.

Nonuniqueness


The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

where m is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

This nonuniqueness leads to a degree of freedom in the formulation of electrodynamics, or gauge freedom, and requires choosing a gauge
Gauge fixing

In the physics of gauge theory, gauge fixing denotes a mathematical procedure for coping with redundant Degrees of freedom in field variables....
.

See also

  • Fundamental theorem of vector analysis
  • Magnetic potential
    Magnetic potential

    The magnetic potential provides a mathematical way to define a magnetic field in classical electromagnetism. It is analogous to the electric potential which defines the electric field in electrostatics....
  • Solenoid
    Solenoid

    A solenoid is a three-dimensional coil. In physics, the term solenoid refers to a loop of wire, often wrapped around a metallic core, which produces a magnetic field when an electric current is passed through it....