Forward modeling the magnetic effects of an inferred source is the basis of magnetic anomaly inversion for estimating subsurface magnetization parameters. This study uses numerical least-squares Gauss–Legendre quadrature (GLQ) integration to evaluate the magnetic potential, anomaly, and gradient components of a cylindrical prism element. Relative to previous studies, it quantifies for the first time the magnetic gradient components, enabling their applications in the interpretation of cylindrical bodies. A comparison of this method to other methods of evaluating the vertical component of the magnetic field associated with a full cylinder shows that it has comparable to improved performance in computational accuracy and speed. Based on the developed theory, a conceptual design is presented for an instrument to measure the magnetic gradient effects of subsurface material in the vicinity of a borehole. The significance of this instrument relative to conventional borehole magnetometers is in its ability to determine the azimuthal directions of magnetic sources within the borehole environment.