About one method of calculation in the arbitrary curvilinear basis of the Laplace operator and curl from the vector function

A simple algorithm for calculating Christoffel symbols, a covariant projection of the result of the Laplace operator's action on the vector, vector curl and other similar operations in an arbitrary oblique base are proposed. For an arbitrary base with ortho e_i is found the expressions of vector projections (ΔA)_i and (rotA)_i, where A is a counter variant vector. Examples of orthonormal bases are considered and general expressions for (ΔA)_i and (rotA)_i for the bases are also given. As a demonstration of the working capacity of the common formulas obtained, detailed calculations of (ΔA)_i and (rotA)_i as an example are made in cases of spherical and cylindrical coordinate systems.