Abstract: According to the definition of the Laplacian operator on Riemannian manifolds，it is a second-order elliptic differential operator obtained by applying the divergence operator to the gradient field of the smooth function on Riemannian manifolds. In this paper，we first specialize Riemannian manifolds into Euclidean spaces， and derive the expression of Laplacian operator in the rectangular coordinate system in the three-dimensional Euclidean space through calculation. And then， by applying transformation formulas between the rectangular coordinate system and the cylindrical coordinate system or the spherical coordinate system，we derive respectively the specific expressions of Laplacian operator in cylindrical and spherical coordinate systems in Euclidean space.

Key words: Laplacian operator, cylindrical coordinate system, spherical coordinate system