Definition:Exterior Derivative

Given an exact $$n \ $$-form $$\omega \ $$ on an $$m \ $$-manifold, local coordinates $$x_1, x_2, \dots, x_m \ $$, and a local coordinate expression for $$\omega \ $$

$$\omega = f(x_1, \dots, x_m) dx_{\phi(1)} \wedge dx_{\phi(2)} \wedge \dots \wedge dx_{\phi(n)} \ $$

where $$\phi:\left\{{1, \dots, n }\right\} \to \left\{{1, \dots, m}\right\} \ $$ is an injection which determines which coordinate vectors $$\omega \ $$ acts on, the exterior derivative $$d\omega \ $$ is the $$(n+1) \ $$-form defined

$$d\omega = \left({ \sum_{k=1}^m \frac{\partial f }{\partial x_k} dx_k }\right) \wedge dx_{\phi(1)} \wedge dx_{\phi(2)} \wedge \dots \wedge dx_{\phi(n)} \ $$

For inexact forms, $$d(a+b) = da + db \ $$