Differential equations as a projection of implicit functions using spatio-temporal Taylor expansion and critical points properties.

This contribution introduces a novel method for formulating differential equations. This method relies on expanding an implicit function that varies with time (denoted as "t") in the space-time domain using Taylor series. This formulation encompasses both ordinary differential equations (ODEs) and partial differential equations (PDEs). In the context of visualizing vector fields, such as fluid flow and electromagnetic fields, the critical points of ODEs play a crucial role in understanding physical phenomena behavior. This paper outlines a general approach for formulating ODEs and PDEs by treating them as time-varying scalar functions using the Taylor expansion. Furthermore, a new condition for identifying critical points is derived and specified specifically for cases where the function is invariant with respect to time (referred to as "t-invariant"). This newly derived formula enhances the detection of critical points, particularly in the context of acquiring and analyzing large 3D fluid flow data. This advancement enables efficient compression of 3D vector data and their representation through radial basis functions (RBFs). [ABSTRACT FROM AUTHOR]