This can be compactly written in terms of the gradient operator ∇, although this final form holds in any curvilinear coordinate system, not just Cartesian.
This expression represents a significant feature of any conservative vector field F, namely F has a corresponding potential ϕ.
[2] Using Stokes' theorem, this is equivalently stated as meaning the curl, denoted ∇×, of the vector field vanishes.
An irrotational flow means the velocity field is conservative, or equivalently the vorticity pseudovector field ω is zero: This allows the velocity potential to be defined simply as: In an electrochemical half-cell, at the interface between the electrolyte (an ionic solution) and the metal electrode, the standard electric potential difference is:[6] where R = gas constant, T = temperature of solution, z = valency of the metal, e = elementary charge, NA = Avogadro constant, and aM+z is the activity of the ions in solution.
[clarification needed] In biology, a potential gradient is the net difference in electric charge across a cell membrane.
However, the Aharonov–Bohm effect is a quantum mechanical effect which illustrates that non-zero electromagnetic potentials along a closed loop (even when the E and B fields are zero everywhere in the region) lead to changes in the phase of the wave function of an electrically charged particle in the region, so the potentials appear to have measurable significance.