Differentiation and integration in functional voxel modeling

This paper presents a simple method for generating the partial derivatives of a multidimensional function using functional voxel models (FV-models). The general principle of constructing, differentiating, and integrating an FV-model is considered for two-dimensional functions. Integration is understood as obtaining local geometrical characteristics for the antiderivative of a local function with solving the Cauchy problem when finally constructing the FV-model. The direct and inverse differentiation algorithm involves the basic properties of the local geometrical characteristics of functional voxel modeling and the inherent linear approximation principle of the codomain of the algebraic function. Simple computer calculations of this algorithm yield an FV-model suitable for any further algebraic operations. An illustrative example of constructing a functional voxel model of a complex two-dimensional algebraic function is provided. Functional voxel models of partial derivatives are obtained based on this model. These models and the boundary condition at a given point are used to obtain an initial FV-model of a complex algebraic function. The approach is applicable to algebraic functions defined on the domain of various dimensions.