Vector Calculus Integral Forms

Abstract

There is an intimate relationship between differential and integral forms in vector calculus (and tensor calculus as well). For example, Maxwell's curl equations for time-varying electric and magnetic field intensities, which are vector differential operators, convert to circulations of these time-varying fields, which are integral forms that describe the electromotive and magnetomotive force (volts and amps), respectively. Further, Maxwell's divergence equations for the electric and magnetic flux densities (differential forms) convert to closed-surface integral forms. These conversion relationships can be developed from a series of theorems from the mathematics of George Green (1828) called Green's identities. Other mathematicians of the 1800s contributed various forms of identities—such as Gauss' and Stokes' theorems, discussed in Sections 5.3 and 5.4, respectively—that significantly add to the tools for converting between differential and integral forms. Since Gauss' work preceded Green's, it would be accurate to describe the relevant Green's forms as generalizations of Gauss'; and since Stokes' theorem followed Green's, one could take the position that Stokes' theorem is a special case of one of Green's identities. Green's mathematics also included the Green's function, which provides an effective method for determining solutions to inhomogeneous differential equations. This process will be covered in Section 5.5; for now, it is sufficient to say that this tool further provides evidence of this differential-integral relationship.