Toroidal coordinates on a "solid limiting bagel" in R³ (i.e. one with a hole shrunk to zero size) are given by

x = (R + r sin(α))cos(θ), y = (R + r sin(α))sin(θ), and z = r cos(α) where R is a fixed constant determining the overall size of the torus, and the three variables r, θ, α are three variables that uniquely determine position within the torus down to the "bagel core," circle which runs through all the centers of the cross-sections. The variable r is the distance of a point from the core. Each level surface r = c, where c is a constant, gives the surface of a torus (donut frosting). θ is the longitudinal angle, which is the same θ as the usual one in cylindrical coordinates. Its level surfaces are the cross sections of the donut cut by a vertical half-plane through the z-axis. α determines the angle around a cross-section one has gone.

This vector field is the derivative of the parameterization with respect to α. and points in the direction of increasing α yielding a field that swirls around the core.