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We have put into correspondence dual bases with direct and reciprocal bases of a curvilinear coordinates system, showing that physics defines geometry.
When dealing with laws of propagation of monochromatic plane waves, the so called "optical transformation theory" in stating an equivalence principle between electromagnetic parameters of a physical medium expressed in Cartesian coordinates and their analogues in vacuum obtained from a curvilinear coordinate transformation, puts in correspondence the Euclidean inhomogeneous and anisotropic physical space, with another homogeneous, isotropic and matter free space, but endowed with a locally flat metric.
Using tensor analysis, which is extremely useful especially for three-dimensional problems in any curvilinear coordinate system, a number of researchers, including Fung [4], Green and Zerna [5] have developed the general theory of elasticity.
The Sinc-Galerkin Method for The Poisson Problem in One Type of Curvilinear Coordinate Domain.
Sokolnikoff (4) and McConnell (5) developed a generalized Hooke's law in curvilinear coordinates.
After presenting the foundations in a form free from any coordinate system, the author follows with a chapter on the technique of writing Navier-Stokes equations and Euler's equations in general steady and nonsteady curvilinear coordinates, as well as the essential aspects of vorticity and stream functions.