Chapter 9: Vector Calculus
Section 9.1 - Student VectorCalculus Package - Overview
Section 9.2 - Vector Objects
Section 9.3 - Differential Operators
Section 9.4 - Differential Identities
Section 9.5 - Line Integrals
Section 9.6 - Surface Integrals
Section 9.7 - Conservative and Solenoidal Fields
Section 9.8 - Divergence Theorem
Section 9.9 - Stokes' Theorem
Section 9.10 - Green's Theorem
Although some use has been made of the Student VectorCalculus package in earlier chapters, this present chapter deals exclusively with the functionalities of the package. Section 9.1 provides an overview of the package, introducing the four basic vector objects the package admits. Vectors defined inside the package carry a coordinate system as an attribute, thereby allowing a rudimentary implementation of differential geometry. These objects are explored in depth in Section 9.2.
Differentiation of these basic objects is considered in Sections 9.3 and 9.4. Various aspects of integration are considered in the six remaining sections of the chapter. Line integrals for work, circulation, and flux through a plane curve appear in Section 9.5; surface integrals and surface flux appear in Section 9.6.
The gradient vector field derives from a scalar by differentiation; finding the scalar when given the gradient is a search for a scalar potential. Finding a vector whose curl is a given vector is a search for a vector potential. Section 9.7 considers these two searches as integration processes, and goes on to consider all the issues connected with conservative and solenoidal vector fields.
The connections between volume, line, and surface integrals are formalized in the Divergence theorem of Section 9.8, Stokes' theorem of Section 9.9, and Green's theorem of Section 9.10.
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