The invariance of charge can be derived as a corollary of Maxwell's equations. The left-hand side of the modified Ampere's law has zero divergence by the div–curl identity. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives: 0 = ∇ ∇ ⋅ ⋅ ( ∇ ∇ × × B ) = ∇ ∇ ⋅ ⋅ ( μ μ 0 ( J + ε ε 0 ∂ ∂ E ∂ ∂ t ) ) = μ μ 0 ( ∇ ∇ ⋅ ⋅ J + ε ε 0 ∂ ∂ ∂ ∂ t ∇ ∇ ⋅ ⋅ E ) = μ μ 0 ( ∇ ∇ ⋅ ⋅ J + ∂ ∂ ρ ρ ∂ ∂ t ) i.e., ∂ ∂ ρ ρ ∂ ∂ t + ∇ ∇ ⋅ ⋅ J = 0. By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary: