The deformable derivative is a recently proposed mathematical tool that interpolates between a function and its classical derivative using a parameterized limit. Initially introduced for single-variable functions, the concept is inherently fractional and has shown promise in various analytical applications. While fractional calculus is a fundamental tool for modeling non-linear, non-local and memory-based systems, it lacks the algebraic simplicity of higher-order derivatives when applied to multivariate calculus. To address this gap, a significant extension of the deformable derivative to multivariable functions is introduced by defining partial deformable derivatives in each coordinate direction while maintaining analytic consistency with classical calculus, and the validation is conducted through several test functions. To further address the lack of higher-order generalizations, the formulation is extended to include second-order deformable derivatives, yielding a higher-order Euler-type theorem. Unlike fractional calculus that relies on complex kernel, the novel method provides a local, computationally efficient identity that preserves the structural elegance of classical homogeneity. Several examples are provided to demonstrate the theoretical results and confirm the smooth convergence of the deformable Euler theorem to its classical counterpart. The proposed framework offers new tools for analysing homogeneous functions and provides a foundation for further extensions in partial differential equations and geometric analysis.
Naidu et al. (Thu,) studied this question.