Abstract
This paper extends the covariant derivative under curved coordinate systems in 3D Euclid space. Based on the axiom of the covariant form invariability, the classical covariant derivative that can only act on components is extended to the generalized covariant derivative that can act on any geometric quantity including base vectors, vectors and tensors. Under the axiom, the algebra structure of the generalized covariant derivative is proved to be covariant differential ring. Based on the powerful operation capabilities and simple analytical properties of the generalized covariant derivative, the tensor analysis in curved coordinate systems is simplified to a large extent.
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The project was supported by the NSFC (11072125 and 11272175), the NSF of Jiangsu Province (SBK201140044), and the Specialized Research Fund for Doctoral Program of Higher Education (20130002110044).
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Yin, YJ. Extension of covariant derivative (I): From component form to objective form. Acta Mech Sin 31, 79–87 (2015). https://doi.org/10.1007/s10409-015-0003-y
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DOI: https://doi.org/10.1007/s10409-015-0003-y