Vector potential

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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

<math> \mathbf{v} = \nabla \times \mathbf{A}. </math>

If a vector field v admits a vector potential A, then from the equality

<math>\nabla \cdot (\nabla \times \mathbf{A}) = 0</math>

(divergence of the curl is zero) one obtains

<math>\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,</math>

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.

1 Theorem
2 Nonuniqueness
3 See also
4 References

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Theorem

Let

<math>\mathbf{v} : \mathbb R^3 \to \mathbb R^3</math>

be solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as |x|→∞. Define

<math> \mathbf{A} (\mathbf{x}) = \frac{1}{4 \pi} \nabla \times \int_{\mathbb R^3} \frac{ \mathbf{v} (\mathbf{y})}{|\mathbf{x} -\mathbf{y} |} \, d\mathbf{y}. </math>

Then, A is a vector potential for v, that is,

<math>\nabla \times \mathbf{A} =\mathbf{v}. </math>

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.

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