Line Integrals, Surface Integrals and Integral Theorem, Lecture notes of Calculus

Download Line Integrals, Surface Integrals and Integral Theorem and more Lecture notes Calculus in PDF only on Docsity!

CHAPTER 10 Line Integrals, Surface Integrals, and Integral Theorems Construction of mathematical models of physical phenomena requires functional domains of greater com- plexity than the previously employed line segments and plane regions. This section makes progress in meet- ing that need by enriching integral theory with the introduction of segments of curves and portions of surfaces as domains. Thus, single integrals as functions defined on curve segments take on new meaning and are then called line integrals. Stokes’s theorem exhibits a striking relation between the line integral of a function on aclosed curve and the double integral of the surface portion that is enclosed. The divergence theorem relates the triple integral of a function on a three-dimensional region of space to its double integral on the bounding surface. ‘The elegant language of vectors best describes these concepts; therefore, it would be useful to reread the introduction to Chapter 7, where the importance of vectors is emphasized. (The integral theorems also are expressed in coordinate form.) Line Integrals The objective of this section is to geometrically view the domain of a vector or scalar function as a segment of a curve. Since the curve is defined on an interval of real numbers, it is possible to refer the function to this primitive domain, but to do so would suppress much geometric insight. dimensional space may be represented by parametric equations: x= AO, y= AM. z=Ala St Sb ied} or in vector notation: x= r(t) (2) where. (see Figure 10.1).