Self-consistent transformation of first-, second-, and third-order potential gradients among Cartesian, cylindrical, and spherical coordinates

Abstract Knowing how to transform potential gradients between different coordinate systems is of fundamental importance in potential field theory. For first- and second-order gradients, such transformations are conventionally dealt with in terms of vector–matrix notation. However, matrix notation is not helpful for deriving the expressions for transformation of third-order potential gradients. In this contribution, we derive the general detailed expressions for transformation of first-, second-, and third-order potential gradients between two coordinate systems by using the direct expansion method. As examples of these general expressions, we derive detailed expressions for forward and inverse transformations of physical components of first-, second-, and third-order potential gradients among Cartesian, cylindrical, and spherical coordinates. Laplace’s equation has been applied for a first validation of partial expressions. However, to validate all newly derived expressions in a systematic way, we propose the closed-loop transformation cycle method that presents a full-fledged commutative diagram of forward and backward transformations among all three coordinate systems, i.e., potential gradients can become themselves after the closed round-trip transformation cycle among Cartesian, cylindrical, and spherical coordinates. Results reveal that this transformation cycle method confirms the correctness of all derived expressions. These general expressions for transformation of first-, second-, and third-order potential gradients can be applied under arbitrary two coordinate systems, and their detailed expressions can be systematically validated by the proposed transformation cycle method.