.
Vector potential
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 negative gradient is a given vector field.
Formally, given a vector field v, a vector potential is a vector field A such that
\( \mathbf{v} = \nabla \times \mathbf{A}. \)
If a vector field v admits a vector potential A, then from the equality
\( \nabla \cdot (\nabla \times \mathbf{A}) = 0 \)
(divergence of the curl is zero) one obtains
\( \nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0, \)
which implies that v must be a solenoidal vector field.
An interesting question is then if any solenoidal vector field admits a vector potential. The answer is yes, if the vector field satisfies certain conditions.[citation needed]
Theorem
Let
\( \mathbf{v} : \mathbb R^3 \to \mathbb R^3 \)
be a solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define
\( \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{\left\|\mathbf{x} -\mathbf{y} \right\|} \, d\mathbf{y}. \)
Then, A is a vector potential for v, that is,
\nabla \times \mathbf{A} =\mathbf{v}. \)
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.
Nonuniqueness
The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is
\( \mathbf{A} + \nabla m \)
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.
See also
Fundamental theorem of vector analysis
Magnetic potential
Solenoid
References
Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
