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In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a '' scalar potential'', 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 of the curl 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. 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. 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 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)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 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 *
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 * 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