| Abstract |
In fluid dynamics we often use orthogonal curvilinear coordinates. For instance the earth coordinates, a version of spherical polar coordinates, make a fundamental framework of oceanography and meteor...ology. The vector representation in such curved coordinates, however, become cumbersome, though brute force calculation is not impossible. In many standard textbooks of fluid dynamics therefore, it seems, only resulting formulas have been described to avoid such a lengthy calculation without geometrical meaning. This short note introduces a notation that applies to general orthogonal curvilinear coordinates and makes it easier to derive curvilinear representation of vector quantities of fluid dynamics in a concise manner. The present notation, I believe, provides us with a more vivid and comprehensible image of vector representation in orthogonal curvilinear coordinates than the traditional one. Also a simple, explicit algorithm is proposed for the derivation of such quantities as divergence, rotation, and Laplacian. Moreover general formulas help us to find intriguing properties pertinent to orthogonal curvilinear coordinates in general. For example the j-th component of the Laplacian of any vector field u has a term proportional to u_j, the coefficient of which must be non-positive.show more
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