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. Vector calculus is a field of mathematics concerned with multivariate real analysis of vectors in 2 or more dimensions. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation about a point. ... A scalar potential is, mathematically, a scalar field whose negative gradient is a given vector field. ... In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. ...

Formally, given a vector field v, a vector potential is a vector field A such that

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

(divergence of the curl is zero) one obtains In vector calculus, the divergence is an operator that measures a vector fields tendency to originate from or converge upon a given point. ... In vector calculus, curl is a vector operator that shows a vector fields rate of rotation about a point. ...

which implies that v must be a solenoidal vector field. This article is in need of attention. ...

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

be solenoidal vector field which is twice continuously differentiable. Assume that v(x) decreases sufficiently fast as |x|→∞. Define This article is in need of attention. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

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

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. The Helmholtz decomposition of a vector field which is twice continuously differentiable, with rapid enough decay at infinity, splits the vector field into a sum of gradient and curl as follows: where represents the Newtonian potential. ... In fluid mechanics, an irrotational vector field is a vector field whose curl is zero. ...

Nonuniqueness

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

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

See also

• solenoid
• fundamental theorem of vector analysis
• magnetic potential

In engineering, a solenoid is a mechanical device that converts energy into linear motion. ... The fundamental theorem of vector analysis states that any vector field meeting certain conditions (of decaying towards infinity) can be resolved into irrotational and solenoidal component vector fields. ... In physics, magnetic potential is a three-dimensional vector field whose curl is the magnetic field in the theory of electromagnetism: In special relativity, the magnetic potential joins with the electric potential into the electromagnetic potential. ...

References

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