Kelvin-Stokes Theorem

Theorem
Let $$S \ $$ be some surface with boundary in $$\R^3 \ $$, and let $$\mathbf{F}:\R^3 \to \R^3 \ $$ be a vector-valued function with Euclidean coordinate expression $$F=f_1\mathbf{i} + f_2\mathbf{j}+f_3\mathbf{k} \ $$, where $$f_i:\R^3 \to \R \ $$. Then

$$\oint_{\partial S} f_1 dx + f_2 dy + f_3 dz = \iint \left({ \nabla \times \mathbf{F}}\right) \cdot \mathbf{n}dA$$

where $$\mathbf{n} \ $$ is the unit normal to $$S \ $$ and $$dA \ $$ is the area element on the surface.