The ∂ ¯ Operator

In an open subset D ⊆ ℝ d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq616.tif"/> with coordinates x ₁,…,x_d a differential form φ of degree r can be written out explicitly in the form Ф = 1 r ! ϕ = ∑ J ϕ j 1 … j r d x j 1 ∧ ⋯ ∧ d x j r https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203750063/1087b710-7222-48dd-8094-26eaf896ae1b/content/eq617.tif"/>