1.NounA differential operator which acts on a differential k-form to yield a differential (k+1)-form, unless the k-form is a pseudoscalar, in which case it yields 0."The exterior derivative of a “scalar”, i.e., a function f=f(x1,x2,...,xn) where the xi’s are coordinates of ℝn, is df=∂f∂x1dx1+∂f∂x2dx2+...+∂f∂xndxn.; The exterior derivative of a k-blade fdxi1∧dxi2∧...∧dxik is df∧dxi1∧dxi2∧...∧dxik.; The exterior derivative d may be though of as a differential operator del wedge: ∇∧, where ∇=∂∂x1dx1+∂∂x2dx2+...+∂∂xndxn. Then the square of the exterior derivative is d2=∇∧∇∧=(∇∧∇)∧=0∧=0 because the wedge product is alternating. (If u is a blade and f a scalar (function), then fu≡f∧u, so d(fu)=∇∧(fu)=∇∧(f∧u)=(∇∧f)∧u=df∧u.) Another way to show that d2=0 is that partial derivatives commute and wedge products of 1-forms anti-commute (so when d2 is applied to a blade then the distributed parts end up canceling to zero.)"