Rectangular pupils are employed in many optical applications such as lasers and anamorphic optics, as well as for detection and metrology systems such as some Shack−Hartmann wavefront sensors and deflectometry systems. For optical fabrication, testing, and analysis in the rectangular domain, it is important to have a well-defined set of polynomials that are orthonormal over a rectangular pupil. Since we often measure the gradient of a wavefront or surface, it is necessary to have a polynomial set that is orthogonal over a rectangular pupil in the vector domain as well. We derive curl (called C) polynomials based on two-dimensional (2-D) versions of Chebyshev polynomials of the first kind. Previous work derived a set of polynomials (called G polynomials) that are obtained from the gradients of the 2-D Chebyshev polynomials. We show how the two sets together can be used as a complete representation of any vector data in the rectangular domain. The curl polynomials themselves or the complete set of G and C polynomials has many interesting applications. Two of those applications shown are systematic error analysis and correction in deflectometry systems and mapping imaging distortion.