Note that the Maxwell Faraday's equation implies that the Electric Field is non conservative when the Magnetic Field varies in time, and is not expressible as the gradient of a scalar field, and not subject to the gradient theorem since its curl is not zero.
note ThAt The maXWEll faraDAY's eQUAtioN imPLies that the ElEctRIc fieLD is noN conSeRvativE wHen The magnetic fieLd varies in time, aND Is nOT EXpREssIble as the grADiEnT oF a scaLar Field, and NOt sUbjECt to the gradIenT tHEorem since iTs CUrl Is NoT zero.