The net current into a volume is I = − − ∬ ∬ S J ⋅ ⋅ d S where S = ∂ V is the boundary of V oriented by outward-pointing normals, and d S is shorthand for N dS, the outward pointing normal of the boundary ∂ V. Here J is the current density (charge per unit area per unit time) at the surface of the volume. The vector points in the direction of the current.