Publisher Summary This chapter describes the exterior derivative of a differential form. The operator d subsumes the ordinary gradient, curl or rotation, and divergence. d is completely independent of coordinate systems. This is clear when d is axiomatized. The inner consistency of the differential form calculus is most important. With a proper formulation, the independence of d on a coordinate system can be obtained as a consequence of the four basic defining properties of the exterior derivative. Vector analysis proves that a curl-free vector field is a gradient by line integrals and a divergence-free vector field is a curl, usually by a brute-force method. The theory of exterior differential forms exposes many types of systems of partial differential equations that are reducible to systems of ordinary differential equations and often solved by quadratures.