Potential, potential field and potential-field gradient data are supplemental to each other for resolving sources of interest in both exploration and solid Earth studies. We propose flexible high-accuracy practical techniques to perform 3D and 2D integral transformations from potential field components to potential and from potential-field gradient components to potential field components in the space domain using cubic B-splines. The spline techniques are applicable to either uniform or non-uniform rectangular grids for the 3D case, and applicable to either regular or irregular grids for the 2D case. The spline-based indefinite integrations can be computed at any point in the computational domain. In our synthetic 3D gravity and magnetic transformation examples, we show that the spline techniques are substantially more accurate than the Fourier transform techniques, and demonstrate that harmonicity is confirmed substantially better for the spline method than the Fourier transform method and that spline-based integration and differentiation are invertible. The cost of the increase in accuracy is an increase in computing time. Our real data examples of 3D transformations show that the spline-based results agree substantially better or better with the observed data than do the Fourier-based results. The spline techniques would therefore be very useful for data quality control through comparisons of the computed and observed components. If certain desired components of the potential field or gradient data are not measured, they can be obtained using the spline-based transformations as alternatives to the Fourier transform techniques.