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Closed forms and exact forms

  1. Let \(\omega\) be a \(k\)-form in an open set \(E\subset R^n\). If there is a \((k-1)\)-form \(\lambda\) in \(E\) such that \(\omega=d\lambda\), then \(\omega\) is said to be exact in \(E\). If \(\omega\) is of class \(\mathscr C'\) and \(d\omega=0\), then \(\omega\) is said to be closed.

  2. If \(\omega\) is of class \(\mathscr C''\) in \(E\), then \[d^2\omega=0\] For a \(0\)-form \(f\in\mathscr C''(E)\) \[\begin{align} d^2f&=d\Biggl(\sum_{j=1}^{n}(D_jf)(\mathbf x)dx_j\Biggr)\\ &=\sum_{j=1}^{n}d(D_jf)(\mathbf x)dx_j\\ &=\sum_{i=1,j=1}^{n}(D_{ij}f)(\mathbf x)dx_i\land dx_j\\ \end{align}\] Since \(D_{ij}f=D_{ji}f\) and \(dx_i\land dx_j=-dx_j\land dx_i\) so \[d^2\omega=(d^2f)\land dx_I=0\] Then every exact form of class \(\mathscr C'\) is closed.

  3. A \(k\)-form \(\omega\) is or not closed can be verified by simply differentiating the coefficients in the standard presentation of \(\omega\). For a \(1\)-form \[\omega=\sum_{i=1}^{n}f_i(\mathbf x)dx_i\] with \(f_i\in\mathscr C'(E)\) for some open set \(E\subset R^n\), is closed if and only if the equations \[(D_jf_i)(\mathbf x)=(D_if_j)(\mathbf x)\] hold for all \(i,j\in\{1,\cdots,n\}\) and for all \(\mathbf x\in E\).

  4. Let \(\omega\) be an exact \(k\)-form in \(E\), then there is a \(k-1\)-form \(\lambda\) in \(E\) with \(d\lambda=\omega\) and Stokes’ theorem asserts that \[\int_\Psi\omega=\int_\Psi d\lambda=\int_{\partial\Psi}\lambda\] for every \(k\)-chain \(\Psi\) of class \(\mathscr C''\) in \(E\).The integral of an exact \(k\)-form in \(E\) is \(0\) over every \(k\)-chain in \(E\) whose boundary is \(0\). All integrals of exact \(1\)-forms in \(E\) are \(0\) over closed curves in \(E\).

  5. Let \(\omega\) be a closed \(k\)-form in \(E\), then \(d\omega=0\) and \[\int_{\partial\Psi}\omega=\int_\Psi d\omega=0\] for every \((k+1)\)-chain \(\Psi\) of class \(\mathscr C''\) in \(E\).

  6. Let \(\Psi\) be a \((k+1)\)-chain in \(E\) and let \(\lambda\) be a \((k-1)\)-form in \(E\), both of class \(\mathscr C''\), and \(d^2\lambda=d\omega=0\), then \[\int_{\partial\partial\Psi}\lambda=\int_{\partial\Psi}d\lambda=\int_{\Psi}d^2\lambda=0\] then \(\partial\partial\Psi=0\), which means the boundary of a boundary is \(0\).

  7. Suppose \(E\) is a convex open set in \(R^n\), \(f\in\mathscr C'(E)\), \(p\) is an integer, \(1\leq p\leq n\), and \[(D_jf)(\mathbf x)=0\quad(p<j\leq n, \mathbf x\in E)\] Then there exist an \(F\in\mathscr C'(E)\) such that \[(D_pF)(\mathbf x)=f(\mathbf x)\] \[(D_jF)(\mathbf x)=0\]

  8. Closed forms are exact in convex sets. If \(E\subset R^n\) is convex and open, if \(\omega\) is a \(k\)-form \((k\ge 1)\) of class \(\mathscr C'\) in \(E\), and if \(d\omega=0\), then there is a \((k-1)\)-form \(\lambda\) in \(E\) such that \(\omega=d\lambda\). For \(p=1,\cdots,n\), let \(Y_p\) denote the set of all \(k\)-forms \(\omega\), of class \(\mathscr C'\) in \(E\), whose standard presentation \[\omega=\sum_{I}f_I(\mathbf x)dx_I \quad I\subset\{1,\cdots,p\}\] if \(f_I(\mathbf x)\ne0\) for some \(\mathbf x\in E\). Assume first that \(\omega\in Y_1\). Then \(\omega=f(\mathbf x)dx_1\). Since \(d\omega=0\), \[(D_jf)(\mathbf x)=0\quad(1<j\leq n, x\in E)\] Then there is an \(F\in \mathscr C'(E)\) such that \(D_1F=f\) and \(D_jF=0,\quad(1<j\leq n)\). Thus \[dF=(D_1F)(\mathbf x)dx_1=f(\mathbf x)dx_1=\omega\] For \(p>1\), make the following hypothesis: Every closed \(k\)-form that belongs to \(Y_{p-1}\) is exact in \(E\). Choose \(\omega\in Y_p\) so that \(d\omega=0\) since \[\omega=\sum_{I}f_I(\mathbf x)dx_I \quad I\subset\{1,\cdots,p\}\] then \[\sum_{I}\sum_{j=1}^{n}(D_jf_I)(\mathbf x)dx_jdx_I=d\omega=0\] Consider a fixed \(j\), with \(p<j\leq n\). Since each \(I\) occurs in \[\omega=\sum_{I}f_I(\mathbf x)dx_I \quad I\subset\{1,\cdots,p\}\] is in \(\{1,\cdots,p\}\), if \(I_1\), \(I_2\) are two of these \(k\)-indices, and if \(I_1\ne I_2\), then the \((k+1)\)-indices \((I_1,j),(I_2,j)\) are distinct. From \[\sum_{I}\sum_{j=1}^{n}(D_jf_I)(\mathbf x)dx_jdx_I=d\omega=0\] we conclude that every coefficient in \[\omega=\sum_{I}f_I(\mathbf x)dx_I\] satisfies \[(D_jf_I)(\mathbf x)=0\quad(\mathbf x\in E, p<j\leq n)\] we can gather those terms in \(\sum_{I}f_I(\mathbf x)dx_I\) that contains \(dx_p\) and rewrite \(\omega\) as \[\omega=a+\sum_{I_0}f_I(\mathbf x)dx_{I_0}\land dx_p\] where \(a\in Y_{p-1}\), each \(I_0\) is an increasing \((k-1)\)-index in \(\{1,\cdots,p-1\}\), and \(I=(I_0,p)\). Since \[(D_jf_I)(\mathbf x)=0\quad(\mathbf x\in E, p<j\leq n)\] there is a function \(F_I\in \mathscr C'(E)\) such that \[D_pF_I=f_I\quad D_jF_I=0\quad(p<j\leq n)\] Define \[\gamma=\omega-(-1)^{k-1}d\Bigl(\sum_{I_0}F_I(\mathbf x)dx_{I_0}\Bigr)\] since \(\sum_{I_0}F_I(\mathbf x)dx_{I_0}\) is a \((k-1)\)-form, it follows that \[\gamma=\omega-\sum_{I_0}\sum_{j=1}^{p}(D_jF_I)(\mathbf x)dx_{I_0}\land dx_j\\ =a-\sum_{I_0}\sum_{j=1}^{p-1}(D_jF_I)(\mathbf x)dx_{I_0}\land dx_j\] which is in \(Y_{p-1}\). Since \(d\omega=0\) and \(d^2\Bigl(\sum_{I_0}F_I(\mathbf x)dx_{I_0}\Bigr)=0\), we have \(d\gamma=0\). Our induction hypothesis shows therefore that \(\gamma=d\mu\) for some \((k-1)\)-form \(\mu\) in \(E\). If \[\lambda=\mu+(-1)^{k-1}\Bigl(\sum_{I_0}F_I(\mathbf x)dx_{I_0}\Bigr)\] we conclude that \(\omega=d\lambda\).