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REMARK There is no doubt that the integral in this definition exists, because $P$ and $Q$ are continuous functions on the compact set $C,$ hence bounded, and ${\gamma}^{\text{'}}$ is integrable, implying that both ${x}^{\text{'}}$ and ${y}^{\text{'}}$ are integrable. Therefore $P\left(\gamma \left(t\right)\right){x}^{\text{'}}\left(t\right)+Q\left(\gamma \left(t\right)\right){y}^{\text{'}}\left(t\right)$ is integrable on $(0,L).$
These differential forms $\omega $ really should be called “differential 1-forms.”For instance, an example of a differential 2-form would look like $R\phantom{\rule{0.166667em}{0ex}}dxdy,$ and in higher dimensions, we could introduce notions of differential forms of higher and higher orders, e.g., in 3 dimension things like $P\phantom{\rule{0.166667em}{0ex}}dxdy+Q\phantom{\rule{0.166667em}{0ex}}dzdy+R\phantom{\rule{0.166667em}{0ex}}dxdz.$ Because we will always be dealing with ${R}^{2},$ we will have no need for higher order differential forms, but the study of such things is wonderful.Take a course in Differential Geometry!
Again, we must see how this quantity ${\int}_{C}\omega $ depends, if it does,on different parameterizations. As usual, it does not.
Suppose $\omega =Pdx+Qdy$ is a differential form on a subset $U$ of ${R}^{2}.$
The simplest interesting example of a differential form is constructed as follows. Suppose $U$ is an open subset of ${R}^{2},$ and let $f:U\to R$ be a differentiable real-valued function of two real variables; i.e., both of its partial derivatives exist at every point $(x,y)\in U.$ (See the last section of Chapter IV.) Define a differential form $\omega =df,$ called the differential of $f,$ by
i.e., $P=tialf/tialx$ and $Q=tialf/tialy.$ These differential forms $df$ are called exact differential forms.
REMARK Not every differential form $\omega $ is exact, i.e., of the form $df.$ Indeed, determining which $\omega $ 's are $df$ 's boils down to what may be the simplest possible partial differential equation problem.If $\omega $ is given by two functions $P$ and $Q,$ then saying that $\omega =df$ amounts to saying that $f$ is a solution of the pair of simultaneous partial differential equations
See part (b) of the exercise below for an example of a nonexact differential form.
Of course if a real-valued function $f$ has continuous partial derivatives of the second order, then [link] tells us that the mixed partials ${f}_{xy}$ and ${f}_{yx}$ must be equal. So, if $\omega =Pdx+Qdy=df$ for some such $f,$ Then $P$ and $Q$ would have to satisfy $tialP/tialy=tialQ/tialx.$ Certainly not every $P$ and $Q$ would satisfy this equation, so it is in fact trivial to find examples of differential forms that are not differentials of functions.A good bit more subtle is the question of whether every differential form $Pdx+Qdy,$ for which $tialP/tialy=tialQ/tialx,$ is equal to some $df.$ Even this is not true in general, as part (c) of the exercise below shows. The open subset $U$ on which the differential form is defined plays a significant role, and, in fact, differential forms provide a way of studyingtopologically different kinds of open sets.
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