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vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
, a vector potential is a vector field whose
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
is a given vector field. This is analogous to a ''
scalar potential In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in trav ...
'', which is a scalar field whose
gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
is a given vector field. Formally, given a vector field v, a ''vector potential'' is a C^2 vector field A such that \mathbf = \nabla \times \mathbf.


Consequence

If a vector field v admits a vector potential A, then from the equality \nabla \cdot (\nabla \times \mathbf) = 0 (
divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of ...
of the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
is zero) one obtains \nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0, which implies that v must be a solenoidal vector field.


Theorem

Let \mathbf : \R^3 \to \R^3 be a solenoidal vector field which is twice
continuously differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
. Assume that decreases at least as fast as 1/\, \mathbf\, for \, \mathbf\, \to \infty . Define \mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf. Then, A is a vector potential for , that is, \nabla \times \mathbf =\mathbf. Here, \nabla_y \times is curl for variable y. Substituting curl '' for the current density j of the retarded potential, you will get this formula. In other words, v corresponds to the
H-field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
. You can restrict the integral domain to any single-connected region Ω. That is, A' below is also a vector potential of v; \mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf. A generalization of this theorem is the Helmholtz decomposition which states that any vector field can be decomposed as a sum of a solenoidal vector field and an irrotational vector field. By
analogy Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...
with Biot-Savart's law, the following \boldsymbol(\textbf) is also qualify as a vector potential for v. :\boldsymbol(\textbf) =\int_\Omega \frac d^3 \boldsymbol Substitute j (
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional a ...
) for v and H (
H-field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to ...
)for A, we will find the Biot-Savart law. Let \textbf\in \mathbb and let the Ω be a star domain centered on the ''p'' then, translating
Poincaré's lemma In mathematics, especially vector calculus and differential topology, a closed form is a differential form ''α'' whose exterior derivative is zero (), and an exact form is a differential form, ''α'', that is the exterior derivative of another dif ...
for differential forms into vector fields world, the followng \boldsymbol(\boldsymbol) is also a vector potential for the \boldsymbol \boldsymbol(\boldsymbol) =\int_0^1 s ((\boldsymbol-\boldsymbol)\times ( \boldsymbol( s \boldsymbol + (1-s) \boldsymbol ))\ ds


Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If is a vector potential for , then so is \mathbf + \nabla f, where f 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.


See also

*
Fundamental theorem of vector calculus 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, rapidly decaying vector field in three dimensions can be resolved int ...
*
Magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic ...
*
Solenoid upright=1.20, An illustration of a solenoid upright=1.20, Magnetic field created by a seven-loop solenoid (cross-sectional view) described using field lines A solenoid () is a type of electromagnet formed by a helix, helical coil of wire whose ...
* Closed and Exact Differential Forms


References

* ''Fundamentals of Engineering Electromagnetics'' by David K. Cheng, Addison-Wesley, 1993. {{Authority control Concepts in physics Potentials Vector calculus Vector physical quantities