# 15.5: Divergence and Curl

- Page ID
- 9063

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One application for divergence occurs in physics, when working with magnetic fields. A magnetic field is a vector field that models the influence of electric currents and magnetic materials. Physicists use divergence in Gauss’s law for magnetism, which states that if \(\vecs{B}\) is a magnetic field, then \(\vecs \nabla \cdot \vecs{B} = 0\); in other words, the divergence of a magnetic field is zero.

## Example \(\PageIndex{2}\): Determining Whether a Field Is Magnetic

Is it possible for \(\vecs{F} (x,y) = \langle x^2 y, \, y - xy^2 \rangle \) to be a magnetic field?

SolutionIf \(\vecs{F}\) were magnetic, then its divergence would be zero. The divergence of \(\vecs{F}\) is

\[\dfrac{\partial}{\partial x} (x^2y) + \dfrac{\partial}{\partial y} (y - xy^2) = 2 xy + 1 - 2 xy = 1 \nonumber \]

and therefore \(\vecs{F}\) cannot model a magnetic field (Figure \(\PageIndex{3}\)).