The Biot-Savart law is a physical law with applications in electromagnetics and, more generally, it is also useful in numerous other applications of vector field analysis. As originally formulated, the law describes the magnetic field set up by a steady current density.

 Mathematically, the Biot-Savart law provides an inverse to the curl operation; the result is unique up to gauge transformation. As such it has numerous additional applications.

The Biot-Savart law is fundamental to magnetostatics just as Coulomb's law is to electrostatics. The Biot-Savart law follows from the Lorentz transformations of the electric field of a point-like electric charge, which results in a magnetic field, and is fully consistent with Ampère's law, much as Coulomb's law is consistent with Gauss' law.

In particular, if we define a differential element of current

then the corresponding differential element of magnetic field is

where