Surface Area

Abstract

In this chapter we shall develop formulas for the m-dimensional area of sets in Rⁿ; and for the area of m-dimensional parametric surfaces ϕ:R^m→ Rⁿ.It is assumed of course that m ≤n.When m = n the theory is just the theory of Lebesgue measure, and the formula for the area of the parametric surface so is just the change of variable formula of the last section (withf = 1):

$$\left| {\varphi \left( \Omega \right)} \right| = \int_\Omega {\left| {\det d\varphi \left( x \right)} \right|dx} $$

(In this case, of course, we do not ordinarily think of area but rather of volume, and we think of so as a solid rather than a surface. When m < n it is more natural to think of surfaces and areas.)