An alternative definition is useful for thermodynamics calculations of the magnetic moment. In this definition, the magnetic dipole moment of a system is the negative gradient of its intrinsic energy, U int, with respect to external magnetic field: m = − − x ^ ^ ∂ ∂ U int ∂ ∂ B x − − y ^ ^ ∂ ∂ U int ∂ ∂ B y − − z ^ ^ ∂ ∂ U int ∂ ∂ B z .