Vector potential

This article is about the general concept in the mathematical theory of vector fields. For the vector potential in electromagnetism, see Magnetic vector potential. For the vector potential in fluid mechanics, see Stream function.

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 v, a vector potential is a $C^{2}$ vector field A such that

$$\mathbf {v} =\nabla \times \mathbf {A} .$$

Consequence

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.

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 at least as fast as $1/\|\mathbf {x} \|$ for $\|\mathbf {x} \|\to \infty$. Define

$$\mathbf {A} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .$$

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

$$\nabla \times \mathbf {A} =\mathbf {v} .$$

Here, $\nabla _{y}\times$ is curl for variable y. Substituting curl[v] 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;

$$\mathbf {A'} (\mathbf {x} )={\frac {1}{4\pi }}\int _{\Omega }{\frac {\nabla _{y}\times \mathbf {v} (\mathbf {y} )}{\left\|\mathbf {x} -\mathbf {y} \right\|}}\,d^{3}\mathbf {y} .$$

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 ${\boldsymbol {A''}}({\textbf {x}})$ is also qualify as a vector potential for v.

${\boldsymbol {A''}}({\textbf {x}})=\int _{\Omega }{\frac {{\boldsymbol {v}}({\boldsymbol {y}})\times ({\boldsymbol {x}}-{\boldsymbol {y}})}{4\pi |{\boldsymbol {x}}-{\boldsymbol {y}}|^{3}}}d^{3}{\boldsymbol {y}}$

Substitute j (current density) for v and H (H-field)for A, we will find the Biot-Savart law.

Let ${\textbf {p}}\in \mathbb {R}$ 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 ${\boldsymbol {A'''}}({\boldsymbol {x}})$ is also a vector potential for the ${\boldsymbol {v}}$

${\boldsymbol {A'''}}({\boldsymbol {x}})=\int _{0}^{1}s(({\boldsymbol {x}}-{\boldsymbol {p}})\times ({\boldsymbol {v}}(s{\boldsymbol {x}}+(1-s){\boldsymbol {p}}))\ ds$

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If A is a vector potential for v, then so is

$$\mathbf {A} +\nabla f,$$

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
• Solenoid
• Closed and Exact Differential Forms

References

• Fundamentals of Engineering Electromagnetics by David K. Cheng, Addison-Wesley, 1993.