We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the $M_1$-integral presented in , using triangle-based partitions.
"Triangle Integral–A Nonabsolute Integration Process Suitable for Piecewise Linear Surfaces." Real Anal. Exchange 36 (2) 373 - 404, 2010/2011.