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Question Number 20058 by Tinkutara last updated on 21/Aug/17

What is the difference between ∮ and ∫? Where is ∮ used?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{difference}\:\mathrm{between}\:\oint\:\mathrm{and} \\ $$$$\int?\:\mathrm{Where}\:\mathrm{is}\:\oint\:\mathrm{used}? \\ $$

Answered by dioph last updated on 21/Aug/17

∮ indicates the integral is calculated in closed path. For instance, for a conservative field A, ∮A ds = 0. Works for any n−dimensional closed surface as well. Gauss′ law for electrostatics states that ∮ D dS = Q, which means that the integral of the electric field density through any closed surface equals the charge encapsulated by the same surface. Hope that helps

$$\oint\:\mathrm{indicates}\:\mathrm{the}\:\mathrm{integral}\:\mathrm{is}\:\mathrm{calculated} \\ $$$$\mathrm{in}\:\:\mathrm{closed}\:\mathrm{path}.\:\mathrm{For}\:\mathrm{instance},\:\mathrm{for} \\ $$$$\mathrm{a}\:\mathrm{conservative}\:\mathrm{field}\:{A},\:\oint{A}\:\mathrm{d}{s}\:=\:\mathrm{0}. \\ $$$$\mathrm{Works}\:\mathrm{for}\:\mathrm{any}\:{n}−\mathrm{dimensional} \\ $$$$\mathrm{closed}\:\mathrm{surface}\:\mathrm{as}\:\mathrm{well}.\:\mathrm{Gauss}'\:\mathrm{law} \\ $$$$\mathrm{for}\:\mathrm{electrostatics}\:\mathrm{states}\:\mathrm{that} \\ $$$$\oint\:{D}\:\mathrm{dS}\:=\:{Q},\:\mathrm{which}\:\mathrm{means}\:\mathrm{that} \\ $$$$\mathrm{the}\:\mathrm{integral}\:\mathrm{of}\:\mathrm{the}\:\mathrm{electric}\:\mathrm{field}\:\mathrm{density} \\ $$$$\mathrm{through}\:\mathrm{any}\:\mathrm{closed}\:\mathrm{surface}\:\mathrm{equals} \\ $$$$\mathrm{the}\:\mathrm{charge}\:\mathrm{encapsulated}\:\mathrm{by}\:\mathrm{the}\:\mathrm{same} \\ $$$$\mathrm{surface}.\:\mathrm{Hope}\:\mathrm{that}\:\mathrm{helps} \\ $$

Commented by Tinkutara last updated on 21/Aug/17

Thank you very much Sir!

$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$

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