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Let

`P(x,y,z),Q(x,y,z)`

, and

`R(x,y,z)`

have continuous first partial derivatives on

`R^(3)`

. Let

`C`

be a closed smooth curve without self intersection, let

`S`

be a smooth surface which can be injectively projected onto the

`xy`

-plane and whose boundary is

`C`

, and let

`hat(n)`

be the unit normal to

`S`

chosen thusly: If when traversing

`C,S`

is on the left, then put

`hat(n)`

on that side. If when traversing

`C,S`

is on the right, then put

`hat(n)`

on the other side. (Think right hand rule.) Use Green's Theorem to prove that

`o\int_C Qdy=∬_(S)((delQ)/(delx)(hat(k))-(delQ)/(delz)(hat(i)))*hat(n)dS`