Equivariant map

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church.

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 vector field A such that

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

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

$\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0$

(divergence of the curl is zero) one obtains

$\mathrm{\nabla }\cdot \mathbf{v}=\mathrm{\nabla }\cdot (\mathrm{\nabla }\times \mathbf{A})=0,$

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 sufficiently fast as ||x||→∞. Define

$\mathbf{A}(\mathbf{x})=\frac{1}{4\pi }\mathrm{\nabla }\times \underset{{\mathbb{ℝ}}^3}{\int }\frac{\mathbf{v}(\mathbf{y})}{\Vert \mathbf{x}-\mathbf{y}\Vert }d\mathbf{y}.$

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

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

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

$\mathbf{A}+\mathrm{\nabla }m$

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, or gauge freedom, and requires choosing a gauge.

See also

• Fundamental theorem of vector analysis
• Magnetic potential
• Solenoid

References

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