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Affine simplexes and chains

  1. A mapping \(\mathbf f\) that carries a vector space \(X\) into a vector space \(Y\) is said to be affine if \[\mathbf f(\mathbf x)-\mathbf f(\mathbf 0)\] is linear, or in other words \[\mathbf f(\mathbf x)-\mathbf f(\mathbf 0)=A\mathbf x\quad A\in L(X,Y)\] The standard simplex \(Q^k\) is defined to be the set of all \(\mathbf u\in R^k\) of the form \[\mathbf u=\sum_{i=1}^{k}a_i\mathbf e_i\] where \(0\leq a_i, \sum a_i\leq 1, i=1,\cdots,k\). Assume that \(\mathbf p_0,\mathbf p_1,\cdots,\mathbf p_k\) are points of \(R^n\), the oriented affine k-simplex \[\sigma=[\mathbf p_0,\mathbf p_1,\cdots,\mathbf p_k]\] is defined to be the k-surface in \(R^n\) with parameter domain \(Q^k\) which is given by the affine mapping \[\sigma(\mathbf u)=\sigma(a_1\mathbf e_1+\cdots+a_k\mathbf e_k)=\mathbf p_0+\sum_{i=1}^{k}a_i(\mathbf p_i-\mathbf p_0)\\ =\mathbf p_0+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf p_1-\mathbf p_0\\ \mathbf p_2-\mathbf p_0\\ \vdots\\ \mathbf p_k-\mathbf p_0\\ \end{bmatrix}\\ =\mathbf p_0+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} A\mathbf e_1\\ A\mathbf e_2\\ \vdots\\ A\mathbf e_k\\ \end{bmatrix}\] \[\sigma(\mathbf 0)=\mathbf p_0, \quad \sigma(\mathbf e_i)=\mathbf p_i\quad (\text{for }1\leq i\leq k)\] and that \[\sigma(\mathbf u)=\mathbf p_0+A\mathbf u\quad(\mathbf u\in Q^k)\] where \(A\in L(R^k, R^n)\) and \[A\mathbf e_i=\mathbf p_i-\mathbf p_0\quad(1\leq i\leq k)\] If \[\overline\sigma=[\mathbf p_{i_0},\mathbf p_{i_1},\cdots,\mathbf p_{{i_k}}]\] where \(\{i_0,i_1,\cdots,i_k\}\) is a permutation of the ordered set \(\{0,1,\cdots,k\}\), we adopt the notation \[\overline\sigma=s(i_0,i_1,\cdots,i_k)\sigma\] thus \[\overline\sigma=\pm\sigma\] Do NOT write \(\overline\sigma=\sigma\) unless they have the same sequence of vertices \(i_0=0, \cdots, i_k=k\).

  2. We write \[\overline\sigma=\varepsilon\sigma\] if \(\varepsilon=1\) we say the two k-simplexes have same orientation, if \(\varepsilon=-1\), we say the two k-simplexes have opposite orientations. \[\sigma(\mathbf u)=\sigma(\sum_{i=1}^{k}a_i\mathbf e_i)=\mathbf p_0+\sum_{i=1}^{k}a_i(\mathbf p_i-\mathbf p_0)=\mathbf p_0+A\sum_{i=1}^{k}a_i\mathbf e_i=\mathbf p_0+A\mathbf u\] If \(A\) is nonsingular and invertible, \(\sigma\) is said to be positively oriented if \(detA=Jacobian(\sigma)>0\). If \(\sigma=\varepsilon\mathbf p_0\quad(\varepsilon=\pm1)\) and if \(f\) is a \(0\)-form (i.e., a real function), we define \[\int_{\sigma}f=\varepsilon f(p_0)\]

  3. If \(\sigma\) is a oriented rectilinear k-simplex in an open set \(E\subset R^n\) and if \(\overline\sigma=\varepsilon\sigma\), if \(f\) is a \(0\)-form, then \[\int_{\sigma}f=\varepsilon f(p_0)\] by definition. If \(f\) is a k-form \(k\ge1\), \(\sigma\) is oriented k-simplex \[\sigma=[\mathbf p_0,\mathbf p_1,\cdots,\mathbf p_k]\] Suppose \(1\leq j\leq k\) and \(\overline\sigma\) is obtained from \(\sigma\) by interchanging \(\mathbf p_0\) and \(\mathbf p_j\), then \(\varepsilon=-1\) and \[\begin{align} \overline\sigma(\mathbf u)&=\mathbf p_j+B\mathbf u\quad(\mathbf u\in Q^k)\\ &=\mathbf p_j+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf p_1-\mathbf p_j\\ \vdots\\ \mathbf p_{j-1}-\mathbf p_j\\ \mathbf p_{0}-\mathbf p_j\\ \mathbf p_{j+1}-\mathbf p_j\\ \vdots\\ \mathbf p_k-\mathbf p_j\\ \end{bmatrix}\\ &=\mathbf p_j+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} B\mathbf e_1\\ \vdots\\ B\mathbf e_{j-1}\\ B\mathbf e_{j}\\ B\mathbf e_{j+1}\\ \vdots\\ B\mathbf e_k\\ \end{bmatrix} \end{align}\] where \(B\) is the linear mapping of \(R^k\) into \(R^n\) defined by \(B\mathbf e_j=\mathbf p_0-\mathbf p_j\), \(B\mathbf e_i=\mathbf p_i-\mathbf p_j\) if \(i\ne j\). Since \[\sigma(\mathbf u)=\sigma(a_1\mathbf e_1+\cdots+a_k\mathbf e_k)=\mathbf p_0+\sum_{i=1}^{k}a_i(\mathbf p_i-\mathbf p_0)\\ =\mathbf p_0+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf p_1-\mathbf p_0\\ \mathbf p_2-\mathbf p_0\\ \vdots\\ \mathbf p_k-\mathbf p_0\\ \end{bmatrix}\\ =\mathbf p_0+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} A\mathbf e_1\\ A\mathbf e_2\\ \vdots\\ A\mathbf e_k\\ \end{bmatrix}\\ =\mathbf p_0+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf x_1\\ \mathbf x_2\\ \vdots\\ \mathbf x_k\\ \end{bmatrix}\quad(\text{If we write }A\mathbf e_i=\mathbf x_i\quad(1\leq i\leq k))\] The \[\begin{align} \overline\sigma(\mathbf u)&=\mathbf p_j+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf p_1-\mathbf p_j\\ \vdots\\ \mathbf p_{j-1}-\mathbf p_j\\ \mathbf p_{0}-\mathbf p_j\\ \mathbf p_{j+1}-\mathbf p_j\\ \vdots\\ \mathbf p_k-\mathbf p_j\\ \end{bmatrix}\\ &=\mathbf p_j+\begin{bmatrix} a_1&a_2&\cdots&a_k\\ \end{bmatrix}\begin{bmatrix} \mathbf x_1-\mathbf x_j\\ \vdots\\ \mathbf x_{j-1}-\mathbf x_j\\ -\mathbf x_j\\ \mathbf x_{j+1}-\mathbf x_j\\ \vdots\\ \mathbf x_k-\mathbf x_j\\ \end{bmatrix} \end{align}\] If we subtract the \(jth\) component \(-\mathbf x_j\) from each of the others, the Jacobians (determinants) of \[\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}\] remain the same and we will obtain component \[\begin{bmatrix} \mathbf x_1\\ \vdots\\ \mathbf x_{j-1}\\ -\mathbf x_j\\ \mathbf x_{j+1}\\ \vdots\\ \mathbf x_k\\ \end{bmatrix}\] which differ from those of \(A\) only in the sign of the \(j{th}\) column. Hence \[\omega(\overline\sigma)=\int_{\overline\sigma}\omega=\int_D\sum a_{i_1\cdots i_k}(\overline\sigma(\mathbf u))\varepsilon\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}d\mathbf u=\varepsilon\int_{\sigma}\omega\] Suppose that \(0<i<j\leq k\) and that \(\overline\sigma\) is obtained from \(\sigma\) by interchanging \(\mathbf p_i\) and \(\mathbf p_j\). Then \[\overline\sigma(\mathbf u)=\mathbf p_0+C\mathbf u\] where \(C\) has the same entries as \(A\). except that the \(i\)th and \(j\)th entries have been interchanged. This implies that \[\omega(\overline\sigma)=\int_{\overline\sigma}\omega=\int_D\sum a_{i_1\cdots i_k}(\overline\sigma(\mathbf u))\varepsilon\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}d\mathbf u=\varepsilon\int_{\sigma}\omega\] with \(\varepsilon=-1\).

  4. An affine k-chain \(\Gamma\) in an open set \(E\subset R^n\) is a collection of finitely many oriented affine k-simplexes \(\sigma_1,\cdots,\sigma_r\) in \(E\). If \(\omega\) is a k-form in \(E\), we define \[\int_{\Gamma}\omega=\sum_{i=1}^{r}\int_{\sigma_i}\omega\] If this equation holds for every k-form \(\omega\) in \(E\), we can write \[\Gamma=\sum_{i=1}^{r}\sigma_i\]

  5. For \(k\ge 1\) the boundary of the oriented affine k-simplex \(\sigma=[\mathbf p_0,\mathbf p_1,\cdots,\mathbf p_k]\) is defined to be the sum of the affine \((k-1)\)-chains with the \(j\)th vertice deleted \[\partial\sigma=\sum_{j=0}^{k}(-1)^j[\mathbf p_0,\cdots,\mathbf p_{j-1},\mathbf p_{j+1},\cdots,\mathbf p_k]\] If \(\sigma=[\mathbf p_0,\mathbf p_1,\mathbf p_2]\) then \[\partial\sigma=[\mathbf p_1,\mathbf p_2]-[\mathbf p_0,\mathbf p_2]+[\mathbf p_0,\mathbf p_1]\] For \(1\leq j\leq k\), the \((k-1)\)-simplex \[\sigma_j=[\mathbf p_0,\cdots,\mathbf p_{j-1},\mathbf p_{j+1},\cdots,\mathbf p_k]\] has \(Q^{k-1}\) as its parameter domain and that it is define by \[\sigma_j(\mathbf u)=\mathbf p_0+B\mathbf u\quad(j=1,2,\cdots,k-1, \mathbf u\in Q^{k-1})\] where \(B\) is the linear mapping from \(R^{k-1}\) to \(R^n\), its columns are determined by \[B\mathbf e_i=\mathbf p_i-\mathbf p_0\quad(1\leq i\leq j-1)\] \[B\mathbf e_i=\mathbf p_{i+1}-\mathbf p_0\quad(j\leq i\leq k-1)\] The simplex \[\sigma_0=[\mathbf p_1,\mathbf p_2,\cdots,\mathbf p_k]\] is given by mapping \[\sigma_0(\mathbf u)=\mathbf p_1+B\mathbf u\] where \(B\mathbf e_i=\mathbf p_{i+1}-\mathbf p_1\quad(1\leq i\leq k-1)\)

  6. Let \(T\) be a \(\mathscr C''\)-mapping of an open set \(E\subset R^n\) into an open set \(V\subset R^m\). \(T\) need not to be \(1-1\). If \(\sigma\) is an oriented affine \(k\)-simplex in \(E\), then the composite mapping \(\Phi=T\circ\sigma\quad(T\sigma \text{ as the simpler form})\) is a \(k\)-surface in \(V\), with parameter domain \(Q^k\). We call \(\Phi\) an oriented \(k\)-simplex of class \(\mathscr C''\). A finit collection \(\Psi\) of oriented \(k\)-simplexes \(\Phi_1,\cdots,\Phi_r\) of class \(\mathscr C''\) in \(V\) is called a \(k\)-chain of class \(\mathscr C''\) in \(V\). If \(\omega\) is a \(k\)-form in \(V\), we define \[\int_{\Psi}\omega=\sum_{i=1}^{r}\int_{\Phi_i}\omega\] and use the notation \(\Psi=\sum\Phi_i\). If \(\Gamma=\sum\sigma_i\) is an affine chain and if \(\Phi_i=T\circ\sigma_i\), we write \[\Psi=T\circ\Gamma=T(\sum\sigma_i)=\sum T\sigma_i\] The boundary of \(\partial(\Phi)\) of the oriented \(k\)-simplex \(\Phi=T\circ\sigma\) is defined to be the \(k-1\) chain \[\partial(\Phi)=\partial(T\circ\sigma)=T(\partial\sigma)\] The boundary \(\partial(\Psi)\) of the \(k\)-chain \(\Psi=\sum\Phi_i\) is defined to be the \((k-1)\) chain \[\partial(\Psi)=\partial(\sum\Phi_i)=\sum\partial\Phi_i\]

  7. Let \(Q^n\) be the standard simplex in \(R^n\), let \(\sigma_0\) be the identity mapping with domain \(Q^n\). Then \(\sigma_0\) can be regarded as a positively oriented \(n\)-simplex in \(R^n\). Its boundary \(\partial\sigma_0\) is an affine \((n-1)\)-chain. This chain is called the positively oriented boundary of the set \(Q^n\). The positively oriented boundary of the set \(Q^3\) is \[[\mathbf e_1,\mathbf e_2,\mathbf e_3]-[\mathbf e_0,\mathbf e_2,\mathbf e_3]+[\mathbf e_0,\mathbf e_1,\mathbf e_3]-[\mathbf e_0,\mathbf e_1,\mathbf e_2]\] Let \(T\) be a \(1-1\) mapping of \(Q^n\) into \(R^n\) of class \(\mathscr C''\) whose Jacobian is positive, let \(E=T(Q^n)\), then \(E\) is the closure of an open subset of \(R^n\). We define the positively oriented boundary of the set \(E\) to be the \((n-1)\)-chain \[\partial E=\partial T=T(\partial\sigma_0)\] Let \[\Omega=E_1\cup\cdots\cup E_r\] where \(E_i=T_i(Q^n)\), each \(T_i\) is a \(1-1\) mapping of \(Q^n\) into \(R^n\) of class \(\mathscr C''\) whose Jacobian is positive, and the interiors of the sets \(E_i\) are pairwise disjoint. Then the \((n-1)\)-chain \[\partial T_1+\cdots+\partial T_r=\partial \Omega\] is called the positively oriented boundary of \(\Omega\).