6.5 Vector fields, differential forms, and line integrals  (Page 3/3)

In fact, although it may seem as if a differential form is really nothing more than a pair of functions,the concept of a differential form is in part a way of organizing our thoughts about partial differential equation problemsinto an abstract mathematical context. This abstraction is a good bit more enlightening in higher dimensional spaces,i.e., in connection with functions of more than two variables. Take a course in Multivariable Analysis!

The next thing we wish to investigate is the continuity of ${\int }_C\omega$ as a function of the curve $C.$ This brings out a significant difference in the concepts of line integrals versis integrals with respect to arc length.For the latter, we typically think of a fixed curve and varying functions, whereas with line integrals, we typically think ofa fixed differential form and variable curves. This is not universally true, but should be kept in mind.

Let $\omega =Pdx+Qdy$ be a fixed, bounded, uniformly continuous differential form on a set $U$ in $R^2,$ and let $C$ be a fixed piecewise smooth curve of finite length $L$ , parameterized by $\phi :[a,b]\to C,$ that is contained in $U.$ Then, given an $ϵ>0$ there exists a $\delta >0$ such that, for any curve $\widehat{C}$ contained in $U,$ $|{\int }_C\omega -{\int }_{\widehat{C}}\omega |<ϵ$ whenever the following conditions on the curve $\widehat{C}$ hold:

1.   $\widehat{C}$ is a piecewise smooth curve of finite length $\widehat{L}$ contained in $U,$ parameterized by $\widehat{\phi }:[a,b]\to \widehat{C}.$
2.   $|\phi \left(t\right)-\widehat{\phi }\left(t\right)|<\delta$ for all $t\in [a,b].$
3.   ${\int }_a^b|{\phi }^{\text{'}}\left(t\right)-{\widehat{\phi }}^{\text{'}}\left(t\right)|dt<\delta .$

Let $ϵ>0$ be given. Because both $P$ and $Q$ are bounded on $U,$ let $M_P$ and $M_Q$ be upper bounds for the functions $\left|P\right|$ and $\left|Q\right|$ respectively. Also, since both $P$ and $Q$ are uniformly continuous on $U,$ there exists a $\delta >0$ such that if $|(c,d)-(c^{\text{'}},d^{\text{'}})|<\delta ,$ then $|P(c,d)-P(c^{\text{'}},d^{\text{'}})|<ϵ/4L$ and $|Q(c,d)-Q(c^{\text{'}},d^{\text{'}})|<ϵ/4L.$ We may also choose this $\delta$ to be less than both $ϵ/4M_P$ and $ϵ/4M_Q.$ Now, suppose $\widehat{C}$ is a curve of finite length $\widehat{L},$ parameterized by $\widehat{\phi }:[a,b]\to \widehat{C},$ and that $|\phi \left(t\right)-\widehat{\phi }\left(t\right)|<\delta$ for all $t\in [a,b],$ and that ${\int }_a^b|{\phi }^{\text{'}}\left(t\right)-{\widehat{\phi }}^{\text{'}}\left(t\right)|<\delta .$ Writing $\phi \left(t\right)=\left(x\right(t),y(t\left)\right)$ and $\widehat{\phi }\left(t\right)=(\widehat{x}\left(t\right),\widehat{y}\left(t\right)),$ we have

as desired.

Again, we have a special notation when the curve $C$ is a graph. If $g:[a,b]\to R$ is a piecewise smooth function, then its graph $C$ is a piecewise smooth curve, and we write ${\int }_{\text{graph}\left(g\right)}Pdx+Qdy$ for the line integral of the differential form $Pdx+Qdy$ over the curve $C=\text{graph}\left(g\right).$

As alluded to earlier, there is a connection between contour integrals and line integrals. It is that a single contourintegral can often be expressed in terms of two line integrals. Here is the precise statement.

Suppose $C$ is a piecewise curve of finite length, and that $f=u+iv$ is a complex-valued, continuous function on $C.$ Let $\phi :[a,b]\to C$ be a parameterization of $C,$ and write $\phi \left(t\right)=x\left(t\right)+iy\left(t\right).$ Then

We just compute:

as asserted.