Let $\mathbf r:\R^2 \to \R^3, \map {\mathbf r} {s, t}$ be a smooth parametrization of $S$ from some region $R$ in the $st$-plane, so that: :$\map {\mathbf r} R = S$ and: :$\map {\mathbf r} {\partial R} = \partial S$ First, we convert the {{LHS}} into a line integral: {{begin-eqn}} {{eqn | l = \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z       | r = \oint_{\partial S} \mathbf F \cdot \rd \mathbf r       | c =  }} {{eqn | r = \oint_{\partial R} \mathbf F \cdot \frac {\partial \mathbf r} {\partial s} \rd s + \mathbf F \cdot \frac {\partial \mathbf r} {\partial t} \rd t       | c =  }} {{end-eqn}} so that if we define: :$\mathbf G = \paren {G_1, G_2} = \paren {\mathbf F \cdot \dfrac {\partial \mathbf r} {\partial s}, \mathbf F \cdot \dfrac {\partial \mathbf r} {\partial t} }$ then: :$\displaystyle \int_{\partial S} \mathbf F \cdot \rd \mathbf r = \int_{\partial R} \mathbf G \cdot \rd \mathbf s$ where $\mathbf s$ is the position vector in the $s t$-plane.  Note that: {{begin-eqn}} {{eqn | l = \frac {\partial G_2} {\partial s}       | r = \mathbf F \cdot \map {\frac \partial {\partial s} } {\frac {\partial \mathbf r} {\partial t} }           + \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}       | c = Derivative of Dot Product of Vector-Valued Functions }} {{eqn | l = \frac {\partial G_1} {\partial t}       | r = \mathbf F \cdot \map {\frac \partial {\partial t} } {\frac {\partial \mathbf r} {\partial s} }           + \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}       | c = Derivative of Dot Product of Vector-Valued Functions }} {{eqn | l = \frac {\partial G_2} {\partial s} - \frac {\partial G_1} {\partial t}       | r = \mathbf F \cdot \map {\frac \partial {\partial s} } {\frac {\partial \mathbf r} {\partial t} }           + \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}       | c = }} {{eqn | o =       | ro= -       | r = \mathbf F \cdot \map {\frac \partial {\partial t} } {\frac {\partial \mathbf r} {\partial s} }           - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}              | c = }} {{eqn | r = \frac {\partial \mathbf r} {\partial t} \cdot \frac {\partial \mathbf F} {\partial s}           - \frac {\partial \mathbf r} {\partial s} \cdot \frac {\partial \mathbf F} {\partial t}       | c = Symmetry of Second Derivatives }} {{end-eqn}} We turn now to the expression on the {{RHS}} and write it in terms of $s$ and $t$: :$\displaystyle \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$ {{begin-eqn}} {{eqn | ro= \displaystyle \ \ \iint_R \Biggl(       | r = \nabla \times \mathbf F \cdot             \paren {\frac {\partial \mathbf r} {\partial s} \times \frac {\partial \mathbf r} {\partial t} }             \Biggr) \rd s \rd t       | c =  }} {{eqn | ro= \displaystyle \ \ \iint_R \Biggl(       | r = \paren {\dfrac {\partial f_3} {\partial y} - \dfrac {\partial f_2} {\partial z} } \mathbf i           + \paren {\dfrac {\partial f_1} {\partial z} - \dfrac {\partial f_3} {\partial x} } \mathbf j           + \paren {\dfrac {\partial f_2} {\partial x} - \dfrac {\partial f_1} {\partial y} } \mathbf k             \Biggr)       | c = Definition:Curl Operator      }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \cdot \Biggl(       | r = \paren {\dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}                   - \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}                    } \mathbf i }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}                   - \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}                    } \mathbf j }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}                   - \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}                    } \mathbf k             \Biggr) \rd s \rd t       | c = Definition:Cross Product }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \paren {\dfrac {\partial f_3} {\partial y}                   - \dfrac {\partial f_2} {\partial z}                    }             \paren {\dfrac {\partial y} {\partial s} \dfrac {\partial z} {\partial t}                   - \dfrac {\partial z} {\partial s} \dfrac {\partial y} {\partial t}                    } }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac {\partial f_1} {\partial z}                   - \dfrac {\partial f_3} {\partial x}                    }             \paren {\dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}                   - \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}                    } }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac {\partial f_2} {\partial x}                   - \dfrac {\partial f_1} {\partial y}                    }             \paren {\dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}                   - \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}                    }             \Biggr) \rd s \rd t      | c = Definition:Dot Product }} {{eqn | o =       | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +        | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}                   }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +        | r = \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}              \Biggr) \rd s \rd t       | c = FOIL }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ -        | r = \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +        | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}             \Biggr) \rd s \rd t       | c = grouping the terms that reference like components of $\mathbf F$ together }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           - \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}                   - \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}                    } }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ -       | r = \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}                   - \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}                    } }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           - \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {\dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}                   - \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}                    }             \Biggr) \rd s \rd t       | c = introducing terms that themselves sum to zero, in order to complete the parts of the derivatives of $\mathbf F$ in a later step }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {- \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}                     - \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}                     - \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}                    } }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {- \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}                     - \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}                     - \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} } }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} }} {{eqn | o =        | ro= \ \ \ \ \ \ \ \ +       | r = \paren {- \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}                     - \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}                     - \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}                    }             \Biggr) \rd s \rd t       | c = grouping terms by sign, but keeping terms that reference like components of $\mathbf F$ together }}  {{eqn | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ - \Biggl(       | r = \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}             \Biggr) \Biggr) \rd s \rd t       | c = grouping all negative terms together and factoring out the negative sign }}  {{eqn | ro= \iint_R \Biggl( \Biggl(       | r = \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_1} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_2} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_3} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t}           + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t} \Biggr) }} {{eqn | o =       | ro= \ \ \ \ \ - \Biggl(       | r = \dfrac {\partial f_1} {\partial x} \dfrac{\partial x}{\partial s} \dfrac{\partial x}{\partial t}           +  \dfrac {\partial f_1} {\partial y} \dfrac{\partial x}{\partial s} \dfrac{\partial y}{\partial t}           +  \dfrac {\partial f_1} {\partial z} \dfrac{\partial x}{\partial s} \dfrac{\partial z}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r = \dfrac {\partial f_2} {\partial x} \dfrac{\partial y}{\partial s} \dfrac{\partial x}{\partial t}           + \dfrac {\partial f_2} {\partial y} \dfrac{\partial y}{\partial s} \dfrac{\partial y}{\partial t}           + \dfrac {\partial f_2} {\partial z} \dfrac{\partial y}{\partial s} \dfrac{\partial z}{\partial t} }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ +       | r  = \dfrac {\partial f_3} {\partial x} \dfrac{\partial z}{\partial s} \dfrac{\partial x}{\partial t}            + \dfrac {\partial f_3} {\partial y} \dfrac{\partial z}{\partial s} \dfrac{\partial y}{\partial t}            + \dfrac {\partial f_3} {\partial z} \dfrac{\partial z}{\partial s} \dfrac{\partial z}{\partial t}              \Biggr)  \Biggr) \rd s \rd t       | c = rearranging into groups, in order to clarify use of the Chain Rule for Real-Valued Functions in the next step }}  {{eqn | ro= \ \ \iint_R \Biggl(       | r = \paren {\dfrac {\partial f_1} {\partial s} \dfrac{\partial x}{\partial t}                   + \dfrac {\partial f_2} {\partial s} \dfrac{\partial y}{\partial t}                   + \dfrac {\partial f_3} {\partial s} \dfrac{\partial z}{\partial t}                    } }} {{eqn | o =       | ro= \ \ \ \ \ \ \ \ -       | r = \paren {\dfrac {\partial f_1} {\partial t} \dfrac{\partial x}{\partial t}                   + \dfrac {\partial f_2} {\partial t} \dfrac{\partial y}{\partial t}                   + \dfrac {\partial f_3} {\partial t} \dfrac{\partial z}{\partial t}                    }           \Biggr) \rd s \rd t       | c = Chain Rule for Real-Valued Functions }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \dfrac {\partial \mathbf F} {\partial s} \cdot \dfrac{\partial \mathbf r}{\partial t}           - \dfrac {\partial \mathbf F} {\partial t} \cdot \dfrac{\partial \mathbf r}{\partial s}             \Biggr) \rd s \rd t       | c = Definition:Dot Product }} {{eqn | ro= \ \ \iint_R \Biggl(       | r = \frac{\partial G_2}{\partial s}           - \frac{\partial G_1}{\partial t}             \Biggr) \rd s \rd t       | c = }} {{end-eqn}} By Green's Theorem, this can be written as: :$\displaystyle \int_{\partial R} \mathbf G \cdot \rd \mathbf s$ Hence both sides of the theorem equation are equal. {{qed}}