in the article Kelvin–Stokes theorem. For a planar surface , a positive path element of curve is defined by the right-hand rule as one that points with the fingers of the right hand when the thumb points in the direction of the normal to the surface . The line integral around is called circulation. A nonzero circulation of is different from the behavior of the electric field generated by static charges. A charge-generated -field can be expressed as the gradient of a scalar field that is a solution to Poisson's equation, and has a zero path integral. See gradient theorem. The integral equation is true for any path through space, and any surface for which that path is a boundary. If the surface is not changing in time, the equation can be rewritten: The surface integral at the right-hand side is the explicit expression for the magnetic flux through . The electric vector field induced by a changing magnetic flux, the solenoidal component of the overall electric field, can be approximated in the non-relativistic limit by the volume integral equation Proof The four Maxwell's equations (including the Maxwell–Faraday equation), along with Lorentz force law, are a sufficient foundation to derive everything in classical electromagnetism. Therefore, it is possible to "prove" Faraday's law starting with these equations. The starting point is the time-derivative of flux through an arbitrary surface (that can be moved or deformed) in space: (by definition). This total time derivative can be evaluated and simplified with the help of the Maxwell–Faraday equation and some vector identities; the details are in the box below: The result is: where is the boundary (loop) of the surface , and is the velocity of a part of the boundary. In the case of a conductive loop, emf (Electromotive Force) is the electromagnetic work done on a unit charge when it has traveled around the loop once, and this work is done by the Lorentz force. Therefore, emf is expressed as where is emf and is the unit charge velocity. In a macroscopic view, for charges on a segment of the loop, consists of two components in average; one is the velocity of the charge along the segment , and the other is the velocity of the segment (the loop is deformed or moved). does not contribute to the work done on the charge since the direction of is same to the direction