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A geodesic on a paraboloid x = sqrt(u)cosv (1) y = sqrt(u)sinv (2) z = u (3) has differential parameters defined by P = ...

The Reynolds transport theorem, also called simply the Reynolds theorem, is an important result in fluid mechanics that's often considered a three-dimensional analog of the ...

A quantity is said to be exact if it has a precise and well-defined value. J. W. Tukey remarked in 1962, "Far better an approximate answer to the right question, which is ...

The great sphere on the surface of a hypersphere is the three-dimensional analog of the great circle on the surface of a sphere. Let 2h be the number of reflecting spheres, ...

Harmonic coordinates satisfy the condition Gamma^lambda=g^(munu)Gamma_(munu)^lambda=0, (1) or equivalently, partial/(partialx^kappa)(sqrt(g)g^(lambdakappa))=0. (2) It is ...

The Lie derivative of tensor T_(ab) with respect to the vector field X is defined by L_XT_(ab)=lim_(deltax->0)(T_(ab)^'(x^')-T_(ab)(x))/(deltax). (1) Explicitly, it is given ...

Written in the notation of partial derivatives, the d'Alembertian square ^2 in a flat spacetime is defined by square ^2=del ^2-1/(c^2)(partial^2)/(partialt^2), where c is the ...

B^^ = T^^xN^^ (1) = (r^'xr^(''))/(|r^'xr^('')|), (2) where the unit tangent vector T and unit "principal" normal vector N are defined by T^^ = (r^'(s))/(|r^'(s)|) (3) N^^ = ...

A coordinate system similar to toroidal coordinates but with fourth-degree instead of second-degree surfaces for constant mu so that the toroids of circular cross section are ...

A divergenceless field can be partitioned into a toroidal and a poloidal part. This separation is important in geo- and heliophysics, and in particular in dynamo theory and ...