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 <math>\mathbf{v}</math>, a vector potential is a <math>C^2</math> vector field <math>\mathbf{A}</math> such that <math display="block"> \mathbf{v} = \nabla \times \mathbf{A}. </math>

Consequence

If a vector field <math>\mathbf{v}</math> admits a vector potential <math>\mathbf{A}</math>, then from the equality <math display="block">\nabla \cdot (\nabla \times \mathbf{A}) = 0</math> (divergence of the curl is zero) one obtains <math display="block">\nabla \cdot \mathbf{v} = \nabla \cdot (\nabla \times \mathbf{A}) = 0,</math> which implies that <math>\mathbf{v}</math> must be a solenoidal vector field.

Theorem

Let <math display="block">\mathbf{v} : \R^3 \to \R^3 </math> be a solenoidal vector field which is twice continuously differentiable. Assume that <math>\mathbf{v}(\mathbf{x})</math> decreases at least as fast as <math> 1/\|\mathbf{x}\| </math> for <math> \| \mathbf{x}\| \to \infty </math>. Define <math display="block"> \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} </math> where <math>\nabla_y \times</math> denotes curl with respect to variable <math>\mathbf{y}</math>. Then <math>\mathbf{A}</math> is a vector potential for <math>\mathbf{v}</math>. That is, <math display="block">\nabla \times \mathbf{A} =\mathbf{v}. </math>

The integral domain can be restricted to any simply connected region <math>\mathbf{\Omega}</math>. That is, <math>\mathbf{A'}</math> also is a vector potential of <math>\mathbf{v}</math>, where <math display="block"> \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}. </math>

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, <math>\mathbf{A}(\mathbf{x})</math> also qualifies as a vector potential for <math>\mathbf{v}</math>, where

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

Substituting <math>\mathbf{j}</math> (current density) for <math>\mathbf{v}</math> and <math>\mathbf{H}</math> (H-field) for <math>\mathbf{A}</math>, yields the Biot-Savart law.

Let <math>\mathbf{\Omega}</math> be a star domain centered at the point <math>\mathbf{p}</math>, where <math>\mathbf{p}\in \R^3</math>. Applying Poincaré's lemma for differential forms to vector fields, then <math>\mathbf{A}(\mathbf{x})</math> also is a vector potential for <math>\mathbf{v}</math>, where

<math>\mathbf{A}(\mathbf{x}) =\int_0^1 s ((\mathbf{x}-\mathbf{p})\times ( \mathbf{v}( s \mathbf{x} + (1-s) \mathbf{p} ))\ ds </math>

Nonuniqueness

The vector potential admitted by a solenoidal field is not unique. If <math>\mathbf{A}</math> is a vector potential for <math>\mathbf{v}</math>, then so is <math display="block"> \mathbf{A} + \nabla f, </math> where <math>f</math> 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
• Solenoidal vector field
• Closed and Exact Differential Forms

References

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