Time Derivatives in Material and Spatial Description—What Are the Differences and Why Do They Concern Us?

  • Elena A. Ivanova
  • Elena N. Vilchevskaya
  • Wolfgang H. Müller
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 60)


This paper has many, albeit mostly didactic objectives. It is an attempt toward clarification of several concepts of continuum theory which can lead and have led to confusion. In a way the paper also creates a bridge between the lingo of the solid mechanics and the fluid mechanics communities. More specifically, an attempt will be made, first, to explain and to interpret the subtleties and the relations between the so-called material and spatial description of continuum fields. Second, the concept of time derivatives in material and spatial description will be investigated meticulously. In particular, it will be explained why and how the so-called material and total time derivatives differ and under which circumstances they turn out to be the same. To that end, material and total time derivatives will be defined separately and evaluated in context with local fields as well as during their use in integral formulations, i.e., when applied to balance equations. As a special example the mass balance is considered for closed as well as open bodies. In the same context the concept of a “moving observation point” will be introduced leading to a generalization of the usual material derivative. When the total time derivative is introduced the distinction between the purely mathematical notion of a coordinate system and the intrinsically physics-based concept of a frame of reference will gain particular importance.


Observation Point Position Vector Material Point Total Derivative Reference Configuration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media Singapore 2016

Authors and Affiliations

  • Elena A. Ivanova
    • 1
    • 2
  • Elena N. Vilchevskaya
    • 1
    • 2
  • Wolfgang H. Müller
    • 3
  1. 1.Peter the Great Saint-Petersburg Plytechnic UniveritySt.-PetersburgRussia
  2. 2.Institute for Problems in Mechanical Engineering, Russian Academy of SciencesSt.-PetersburgRussia
  3. 3.Chair of Continuum Mechanics and Materials TheoryInstitute of Mechanics, Technical University of BerlinBerlinGermany

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