ABSTRACT
In Chapter 4, scalar and vector fields were defined and the properties of gradient, divergence, and curl were discussed; and in Chapter 5 we were concerned with various integrals of scalar and vector fields, and the techniques whereby such integrals are evaluated. The ground has thus been prepared for the two central theorems in vector analysis: (i) the divergence theorem (also called Gauss’s theorem), which relates the integral of a vector field F over a closed surface S to the volume integral of div F over the region bounded by S; and (ii) Stokes’s theorem which relates the integral of a vector field F around a closed curve https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734414/0eb74bf2-ffd1-4299-955f-ba87c452e44a/content/commn6_221.tif"/> to the integral of curl F over any open surface S bounded by https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203734414/0eb74bf2-ffd1-4299-955f-ba87c452e44a/content/commn6_221.tif"/>. In this chapter we shall prove these theorems and some related results.
