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5 Vector Integral Calculus

Part I. Two-Dimensional Theory

5.1 INTRODUCTION

The topic of this chapter is line and surface integrals. It will be seen that these can both be regarded as integrals of vectors and that the principal theorems can be most simply stated in terms of vectors; hence the title “vector integral calculus.” A familiar line integral is that of arc length: $\underset{C}{\int }ds.$ The subscript C indicates that one is measuring the length of a curve C, as in Fig. $5.1.$ If C is given in parametric form $x\phantom{\rule{0ex}{0ex}}=x\left(t\right),\phantom{\rule{0ex}{0ex}}y\phantom{\rule{0ex}{0ex}}=y\left(t\right),\phantom{\rule{0ex}{0ex}}$ the line integral reduces to the ordinary definite integral:

$\underset{C}{\int }ds\phantom{\rule{0ex}{0ex}}={\int }_{{t}_{1}}^{{t}_{2}}\sqrt{{\left(\frac{dx}{dt}\right)}^{2}+{\left(\frac{dy}{dt}\right)}^{2}}dt.$

If the curve C represents a wire whose density (mass per unit length) varies
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along C, then the wire has a total                   (Braille page no. a279)
mass $M\phantom{\rule{0ex}{0ex}}=\underset{C}{\int }f\left(x,\phantom{\rule{0ex}{0ex}}y\right)ds,\phantom{\rule{0ex}{0ex}}$ where $f\left(x,\phantom{\rule{0ex}{0ex}}y\right)$ is the density at the point $\left(x,\phantom{\rule{0ex}{0ex}}y\right)$ of the wire. The new integral can be expressed in terms of a parameter as previously or can be thought of simply as a limit of a sum $\underset{C}{\int }f\left(x,\phantom{\rule{0ex}{0ex}}y\right)ds\phantom{\rule{0ex}{0ex}}=\mathrm{lim}\underset{i\phantom{\rule{0ex}{0ex}}=1}{\overset{n}{\Sigma }}f\left({x}_{i}^{*},\phantom{\rule{0ex}{0ex}}{y}_{i}^{*}\right){\Delta }_{i}s.$
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Here the curve has been subdivided into n pieces of lengths ${\Delta }_{1}s,\phantom{\rule{0ex}{0ex}}{\Delta }_{2}s,\phantom{\rule{0ex}{0ex}}\dots ,\phantom{\rule{0ex}{0ex}}{\Delta }_{n}s,\phantom{\rule{0ex}{0ex}}$ and the point $\left({x}_{i}^{*},\phantom{\rule{0ex}{0ex}}{y}_{i}^{*}\right)$ lies on the ith piece. The limit is taken as n becomes infinite, while the maximum ${\Delta }_{i}s$ approaches $0.$