Potential-field and gradient-component transformations and derivative computations are necessary for many techniques of data enhancement, direct interpretation, and inversion. We advance new unified formulas for fast interpolation, differentiation, and integration and propose flexible high-precision algorithms to perform 3D and 2D potential-field- and gradient component transformations and derivative computations in the space domain using cubic B-splines. The spline-based algorithms are applicable to uniform or nonuniform rectangular grids for the 3D case and to regular or irregular grids for the 2D case. The fast Fourier transform (FFT) techniques require uniform grid spacing. The spline-based horizontal-derivative computations can be done at any point in the computational domain, whereas the FFT methods use only grid points. Comparisons between spline and FFT techniques through two gravity-gradient examples and one magnetic example show that results computed with the spline technique agree better with the exact theoretical data than do results computed with the FFT technique.