On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan; it allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a k-form is thought of as measuring the flux through an infinitesimal k-parallelepiped, then its exterior derivative can be thought of as measuring the net flux through the boundary of a (k+1)-parallelepiped. Definition T...

In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus: :(f\cdot g)'=f'\cdot g+f\cdot g' \,\! . or in the Leibniz notation thus: :\dfrac{d}{dx}(u\cdot v)=u\cdot \dfrac{dv}{dx}+v\cdot \dfrac{du}{dx}. In the notation of differentials this can be written as follows: : d(uv)=u\,dv+v\,du. The derivative of the product of three functions is: :\dfrac{d}{dx}(u\cdot v \cdot w)=\dfrac{du}{dx} \cdot v \cdot w + u \cdot \dfrac{dv}{dx} \cdot w + u\cdot v\cdot \dfrac{dw}{dx}. Discovery Discovery of this rule is credi...