On the material time derivative of volume, surface, and line integrals

TL;DR

This paper presents a purely mathematical approach to deriving the material time derivative of volume, surface, and line integrals, avoiding reliance on referential configurations often used in continuum mechanics.

Contribution

It introduces a formal, physics-independent method for calculating the material time derivative of integrals, removing the need for referential configurations.

Findings

Provides a new mathematical derivation method

Eliminates dependence on referential configurations

Simplifies the understanding of integral derivatives in continuum mechanics

Abstract

Traditional derivation of the material time derivative of volume, surface, and line integrals relies upon the notion of a referential configuration of continuum. Such a notion, however, is artificial and, probably, somewhat misleading in cases of liquids, gases, plastic flow of solids etc. It is, consequently, desirable to separate the formal calculation of the material time derivative of time-dependent integrals from any physics that can be related with them. Such a separation is targeted in the present letter where no referential continuum configuration or other physical notion is involved. The material time derivative of volume, surface, and line integrals is presented as a purely mathematical manipulation.