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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 negative gradient is a given vector field. Formally, given a vector field v, a vector potential is a vector field A such that abla imes mathbf. If a vector field v admits a vector potential A, then from the equality abla cdot ( abla imes mathbf) = 0 (divergence of the curl is zero) one obtains abla cdot mathbf = abla cdot ( abla imes mathbf) = 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 affirmative, if the vector potential satisfies certain conditions.
Theorem Let be solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as ||x||→∞. Define abla imes int_ rac , dmathbf. Then, A is a vector potential for v, that is, abla imes mathbf =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. Nonuniqueness The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is abla 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, and is referred to as choosing a gauge. See also | ||||||||
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