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:

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
$$M\phantom{\rule{0ex}{0ex}}=\underset{C}{\int}f(x,\phantom{\rule{0ex}{0ex}}y)ds,\phantom{\rule{0ex}{0ex}}$$
where $f(x,\phantom{\rule{0ex}{0ex}}y)$ is the density at the point $(x,\phantom{\rule{0ex}{0ex}}y)$
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
$$\underset{C}{\int}f(x,\phantom{\rule{0ex}{0ex}}y)ds\phantom{\rule{0ex}{0ex}}=\mathrm{lim}\underset{i\phantom{\rule{0ex}{0ex}}=1}{\overset{n}{\Sigma}}f({x}_{i}^{*},\phantom{\rule{0ex}{0ex}}{y}_{i}^{*}){\Delta}_{i}s.$$

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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
$({x}_{i}^{*},\phantom{\rule{0ex}{0ex}}{y}_{i}^{*})$ lies on the *i*th piece.
The limit is taken as `n` becomes infinite,
while the maximum ${\Delta}_{i}s$ approaches $0.$