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

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

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

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

(divergence of the curl is zero) one obtains

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

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.

Theorem

---

Let

$\backslash mathbf\{v\}\; :\; \backslash mathbb\; R^3\; \backslash to\; \backslash mathbb\; R^3$

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

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

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

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

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

$\backslash mathbf\{A\}\; +\; \backslash nabla\; m$

where m is any continuously differentiable scalar function. This follows from the fact that the curl of the gradient is zero.

See also

---

• Fundamental theorem of vector analysis
• Magnetic potential
• Solenoid

References

---

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

Potential | Vector calculus

Potencial vectorial | Vektorpotential