▁electric ▁potential , ▁field , ▁and ▁perm itt ivity , ▁respectively , ▁, ▁ ▁and ▁ ▁are ▁the ▁condu ction , ▁disp lacement ▁and , ▁in ▁a ▁' qu asi - elect ro static ' ▁assumption , ▁total ▁current ▁density , ▁respectively ; ▁ ▁is ▁an ▁arbitrary ▁ir rot ational ▁vector ▁in ▁an ▁arbitrary ▁volume ▁ ▁en closed ▁by ▁the ▁surface ▁S , ▁with ▁the ▁constraint ▁that ▁. ▁Now ▁let ▁us ▁integrate ▁over ▁ ▁the ▁scalar ▁product ▁of ▁the ▁vector ▁ ▁by ▁the ▁two ▁members ▁of ▁the ▁above - ment ioned ▁current ▁equation . ▁Indeed , ▁by ▁applying ▁the ▁diver gence ▁theorem , ▁the ▁vector ▁identity ▁, ▁the ▁above - ment ioned ▁constraint ▁and ▁the ▁fact ▁that ▁, ▁we ▁obtain ▁the ▁electro kin emat ics ▁theorem ▁in ▁the ▁first ▁form ▁▁ ▁, ▁ ▁which , ▁taking ▁into ▁account ▁the ▁cor pus cular ▁nature ▁of ▁the ▁current ▁, ▁where ▁ ▁is ▁the ▁Di rac ▁delta ▁function ▁and ▁N ( t ) ▁is ▁the ▁car rier ▁number ▁in ▁ ▁at ▁the ▁time ▁t , ▁becomes ▁▁ ▁. ▁ ▁A ▁component ▁ ▁of ▁the ▁total ▁electric ▁potential ▁ ▁is ▁due ▁to ▁the ▁voltage ▁ ▁applied ▁to ▁the ▁k th ▁electro de ▁on ▁S , ▁on ▁which ▁ ▁( and ▁with ▁the ▁other ▁boundary ▁conditions ▁ ▁on ▁the ▁other ▁electro des ▁and ▁for ▁), ▁and ▁each ▁component ▁ ▁is ▁due ▁to ▁the ▁j th ▁charge ▁car rier ▁q j ▁, ▁being ▁ ▁for ▁ ▁and ▁ ▁over ▁any ▁electro de ▁and ▁for ▁. ▁Moreover , ▁let ▁the