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Surface integrals of differential 2-forms

Let

be a differential 2-form defined on a surface S, and let

be an orientation preserving parametrization of S with (s, t) {\displaystyle (s,t)} in D. Changing coordinates from (x, y) {\displaystyle (x,y)} to (s, t) {\displaystyle (s,t)}, the differential forms transform as

So d x d y {\displaystyle \mathrm {d} x\mathrm {d} y} transforms to ∂ (x, y) ∂ (s, t) d s d t {\displaystyle {\frac {\partial (x,y)}{\partial (s,t)}}\mathrm {d} s\mathrm {d} t}, where ∂ (x, y) ∂ (s, t) {\displaystyle {\frac {\partial (x,y)}{\partial (s,t)}}} denotes the determinant of the Jacobian of the transition function from (s, t) {\displaystyle (s,t)} to (x, y) {\displaystyle (x,y)}. The transformation of the other forms are similar.

Then, the surface integral of f on S is given by

where

is the surface element normal to S.

Let us note that the surface integral of this 2-form is the same as the surface integral of the vector field which has as components f x {\displaystyle f_{x}}, f y {\displaystyle f_{y}} and f z {\displaystyle f_{z}}.