A typical example where the straight-line distance between two points is converted to curvilinear coordinates can be found in the applications of Legendre polynomials in physics.
42.
In mathematics, the "'directional derivative "'of a multivariate differentiable function along a given curvilinear coordinate curves, all other coordinates being constant.
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It is possible to write the quantum mechanical kinetic energy operator for such curvilinear coordinates, but it is hard to formulate a general theory applicable to any molecule.
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There are 11 orthogonal curvilinear coordinate systems ( to within a conformal symmetry ) in which the 3-variable Helmholtz equation can be solved using the separation of variables technique.
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There are 17 orthogonal curvilinear coordinate systems ( to within a conformal symmetry ) in which the 3-variable Laplace equation can be solved using the separation of variables technique.
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More general curvilinear coordinates could be used instead, and are in contexts like continuum mechanics and general relativity ( in the latter case one needs an additional time coordinate ).
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*PM : derivation of unit vectors in curvilinear coordinates, id = 9321 new !-- WP guess : derivation of unit vectors in curvilinear coordinates-- Status:
48.
*PM : derivation of unit vectors in curvilinear coordinates, id = 9321 new !-- WP guess : derivation of unit vectors in curvilinear coordinates-- Status:
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Is the kinetic energy of the particle, and " g bc " the covariant components of the " metric tensor " of the curvilinear coordinate system.
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They are not in general fixed directions in space, which happens to be the case for simple Cartesian coordinates, and thus there is generally no natural global basis for curvilinear coordinates.
