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Third-order gradients of the gravitational potential (gravitational curvatures) have already found some applications in geosciences. Observability of these parameters, describing the Earth's gravitational field in a more complex way than any other currently available gravitational parameter, such as gravitational acceleration (first-order gradient) or gravitational (second-order) gradient, is currently discussed by physicists. Moreover, first designs of observational devices (sensors) have already been proposed. The spherical harmonic analysis and synthesis are the common tools used by geoscientists to study spectral properties of various functionals of the Earth's gravitational potential. However, the conventional spherical harmonic expansions of the gravitational curvatures in the local north-oriented reference frame have rather complicated forms that depend on the first-, second- and third-order derivatives of the associated Legendre functions. Moreover, some of these expansions also contain singular terms at the poles. In this paper, the conventional series are transformed to new simpler and non-singular forms based on relations between the associated Legendre functions and their derivatives. Numerical experiments demonstrate the applicability and correctness of the new expressions. [ABSTRACT FROM AUTHOR]