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 gradient is a given vector field. Formally, given a vector field \mathbf{v} , a vector potential is a C^2 vector field \mathbf{A} such that \mathbf{v} = \nabla \times \mathbf{A}.

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 is a given vector field. Formally, given a vector field \mathbf{v} , a vector potential is a C^2 vector field \mathbf{A} such that \mathbf{v} = \nabla \times \mathbf{A}. ==Consequence== If a vector field \mathbf{v} admits a vector potential \mathbf{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 \mathbf{v} must be a solenoidal vector field. ==Theorem== Let \mathbf{v} : \R^3 \to \R^3 be a solenoidal vector field which is twice continuously differentiable. Assume that \mathbf{v}(\mathbf{x}) decreases at least as fast as 1/\|\mathbf{x}\| for \| \mathbf{x}\| \to \infty . Define \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\mathbb R^3} \frac{ \nabla_\mathbf{s} \times \mathbf{v} (\mathbf{s})}{\left\|\mathbf{x} -\mathbf{s} \right\|} \, d^3\mathbf{s} where \nabla_\mathbf{s} \times denotes curl with respect to variable \mathbf{s} . Then \mathbf{A} is a vector potential for \mathbf{v} . That is, \nabla \times \mathbf{A} =\mathbf{v}. The integral domain can be restricted to any simply connected region \Omega . That is, \mathbf{A'} also is a vector potential of \mathbf{v} , where \mathbf{A'} (\mathbf{x}) = \frac{1}{4 \pi} \int_{\Omega} \frac{ \nabla_\mathbf{s} \times \mathbf{v} (\mathbf{s})}{\left\|\mathbf{x} -\mathbf{s} \right\|} \, d^3\mathbf{s}. A generalization of this theorem is the Helmholtz decomposition theorem, 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 the Biot–Savart law, \mathbf{A}(\mathbf{x}) also qualifies as a vector potential for where \mathbf{A}(\mathbf{x}) = \int_\Omega \frac{\mathbf{v}(\mathbf{s}) \times (\mathbf{x} - \mathbf{s})}{4 \pi \left|\mathbf{x} - \mathbf{s}\right|^3} d^3 \mathbf{s} Substituting \mathbf{j} (current density) for \mathbf{v} and \mathbf{H} (H-field) for \mathbf{A} , yields the Biot–Savart law. Let \Omega be a star domain centered at the point \mathbf{p} , where \mathbf{p}\in \R^3 . Applying Poincaré's lemma for differential forms to vector fields, then \mathbf{A}(\mathbf{x}) also is a vector potential for \mathbf{v} , where \mathbf{A}(\mathbf{x}) =\int_0^1 s \left[(\mathbf{x}-\mathbf{p})\times \mathbf{v}( s \mathbf{x} + (1{-}s) \mathbf{p})\right] ds ==Nonuniqueness== The vector potential admitted by a solenoidal field is not unique. If \mathbf{A} is a vector potential for \mathbf{v} , then so is \mathbf{A} + \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.

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