Generalized forms and vector fields

TL;DR

This paper introduces a generalized framework for vector fields and forms on manifolds, defining new operations like the generalized Lie derivative and commutator, and explores their identities and applications.

Contribution

It develops a novel generalized vector and form calculus on manifolds, extending classical concepts with new definitions and identities.

Findings

Defined generalized vectors and forms on manifolds.

Introduced generalized Lie derivative and commutator.

Explored identities and an application of the generalized calculus.

Abstract

The generalized vector is defined on an $n$ dimensional manifold. Interior product, Lie derivative acting on generalized $p$-forms, $-1\le p\le n$ are introduced. Generalized commutator of two generalized vectors are defined. Adding a correction term to Cartan's formula the generalized Lie derivative's action on a generalized vector field is defined. We explore various identities of the generalized Lie derivative with respect to generalized vector fields, and discuss an application.