Stokes' Theorem

= Classical Stokes' Theorem in Three Dimensions =

Theorem
$$\oint_{\partial S} f_1 dx + f_2 dy + f_3 dz = \iint \hat{n} \cdot \operatorname{curl} \hat{F} dA$$

Proof
= Generalized Stokes' Theorem =

Theorem
If $$\omega$$ is any smooth (k-1) form on a smooth manifold X, then

$$\int_{\partial X} \omega = \int_X d \omega$$

where $$d\omega \ $$ is the exterior derivative of $$\omega \ $$.