The Berry curvature is an anti-symmetric second-rank tensor derived from the Berry connection via Ω Ω n , μ μ ν ν ( R ) = ∂ ∂ ∂ ∂ R μ μ A n , ν ν ( R ) − − ∂ ∂ ∂ ∂ R ν ν A n , μ μ ( R ) . In a three-dimensional parameter space the Berry curvature can be written in the pseudovector form Ω Ω n ( R ) = ∇ ∇ R × × A n ( R ) . The tensor and pseudovector forms of the Berry curvature are related to each other through the Levi-Civita antisymmetric tensor as Ω Ω n , μ μ ν ν = ϵ ϵ μ μ ν ν ξ ξ Ω Ω n , ξ ξ . In contrast to the Berry connection, which is physical only after integrating around a closed path, the Berry curvature is a gauge-invariant local manifestation of the geometric properties of the wavefunctions in the parameter space, and has proven to be an essential physical ingredient for understanding a variety of electronic properties.