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SUBSCRIPTIONS & PRICING
GENERAL INFORMATION
chapter 5, Vector Calculus Integral Forms
Table of Contents
- 1. Introduction
Chapter Contents
- 5.1 Line Integrals of Vector (and Other Tensor) Fields
- 5.1.1 Line integrals of scalar, vector, and tensor fields with dot-, cross-, and direct-product integrands
- 5.1.2 Examples of form (5.1-1): Line integral of the tangential component of
along path L - (a) Examples in mechanics—force and work
- (b) Electrostatics—electric field intensity and electric potential
- (c) Path dependence of tangential line integrals
- 5.1.3 Other line-integral examples
- 5.2 Surface Integrals of Vector (and Other Tensor) Fields
- 5.2.1 Surface integrals of scalar, vector and other tensor fields with dot-, cross-, and tensor-product integrands
- 5.2.2 Surface integral applications
- 5.3 Gauss' (Divergence) Theorem
- 5.3.1 Gauss' law
- 5.3.2 Derivation of Gauss' divergence theorem
- 5.3.3 Implications of divergence theorem on the source distribution
- 5.3.4 Application: The energy in electromagnetic fields—Pointing's theorem
- 5.4 Stokes' (Curl) Theorem
- 5.4.1 Ampere's circuital law
- 5.4.2 Derivation of Stokes' theorem
- 5.4.3 Implications of Stokes' theorem
- 5.5 Green's Mathematics
- 5.5.1 Green's identities
- 5.5.2 Green's function
- 5.5.3 Applications of Green's mathematics
- (a) Retarded electric scalar potential
- (b) Retarded magnetic vector potential
- References
Excerpt
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.
Before probing into the powerful mathematics of these forms and theorems, we first elaborate on line and surface integrals for two reasons. First, all of the aforementioned theorems involve line or surface integrals or both. Secondly, this elaboration will provide comprehensiveness so that the breadth of physical applications may be described. In Section 2.4, line and surface integrals were introduced as examples of integrands made up of the vector dot product of a vector field with the vector line and surface differentials 
and 
[see Eqs. (2.4–20)–(2.4–23)]. In Sections 5.1 and 5.2, line and surface integrals, respectively, are covered more generally.
©2005 Society of Photo-Optical Instrumentation Engineers
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