Conveniently, the second term vanishes in the difference, by equality of mixed partials. So, ∂ ∂ P v ∂ ∂ u − − ∂ ∂ P u ∂ ∂ v = ∂ ∂ ( F ∘ ∘ ψ ψ ) ∂ ∂ u ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ v − − ∂ ∂ ( F ∘ ∘ ψ ψ ) ∂ ∂ v ⋅ ⋅ ∂ ∂ ψ ψ ∂ ∂ u = ∂ ∂ ψ ψ ∂ ∂ v ⋅ ⋅ ( J ψ ψ ( u , v ) F ) ∂ ∂ ψ ψ ∂ ∂ u − − ∂ ∂ ψ ψ ∂ ∂ u ⋅ ⋅ ( J ψ ψ ( u , v ) F ) ∂ ∂ ψ ψ ∂ ∂ v (chain rule) = ∂ ∂ ψ ψ ∂ ∂ v ⋅ ⋅ ( J ψ ψ ( u , v ) F − − ( J ψ ψ ( u , v ) F ) T ) ∂ ∂ ψ ψ ∂ ∂ u