Suppose \(I^k\) is a k-cell in \(R^k\) consisting of all \[\mathbf x=(x_1,\cdots,x_k)\] such that \(a_i\leq x_i\leq b_i\quad(i=1,\cdots,k)\) and \(f\) is a real continuous function on \(I^k\). Put \(f=f_k\) and define \[f_{k-1}(x_1,\cdots,x_{k-1})=\int_{a_k}^{b_k}f_{k}(x_1,\cdots,x_{k-1},x_k)dx_k\] We repeat this process \(k\) steps and obtain a function, which is defined \[L(f)=\int_{I^k}f(\mathbf x)d\mathbf x\] or \[\int_{I^k}f\] If \(L'(f)\) is the result obtained by carrying out the \(K\) integration in some other order, then for every \(f\in\mathscr C(I^k), L(f)=L'(f)\).
The support of a function \(f\) on \(R^k\) is the closure of the set of all points \(\mathbf x\in R^k\) at which \(f(\mathbf x)\ne0\), if \(f\) is a continuous function with compact support, let \(I^k\) be any k-cell which contains the support of \(f\), and define \[\int_{R^k}f=\int_{I^k}f\]
If \(\mathbf G\) maps an open set \(E\subset R^n\) and if there is an integer \(m\) and a real function \(g\) with domain \(E\) such that \[\mathbf G(\mathbf x)=\sum_{i\ne m}^{n}x_i\mathbf e_i+g(\mathbf x)\mathbf e_m\quad(\mathbf x\in E)\] or \[\mathbf G(\mathbf x)=\mathbf x+[g(\mathbf x)-x_m]\mathbf e_m\] then function \(\mathbf G\) is called primitive, a primitive mapping is thus that changes at most one coordinate. If \(g\) is differentiable at some point \(\mathbf a\in E\), so is \(\mathbf G\). The matrix of the operator \(\mathbf G'(\mathbf a)\) has \[(D_1g_m)(\mathbf a),\cdots,(D_mg_m)(\mathbf a),\cdots,(D_ng_m)(\mathbf a)\] as its \(mth\) row. For \(j\ne m\), we have \(D_jg_j=1\) and \(D_ig_j=0\) if \(i\ne j\). The Jacobian of \(\mathbf G\) at \(\mathbf a\) is thus given by \[J_{\mathbf G}(\mathbf a)=\text{det}[\mathbf G'(\mathbf a)]=(D_mg_m)(\mathbf a)\], so \(\mathbf G'(\mathbf a)\) is invertible if and only if \((D_mg_m)(\mathbf a)\ne 0\). The primitive is multiplication each element in a row of the matrix by a non-zero number.
A linear operator \(\mathbf B\) on \(R^n\) that interchanges some pair of members of the standard basis and leaves the others fixed will be called a flip. The flip is the interchange of two rows of the matrix. The projections \(P_0,\cdots,P_n\) in \(R^n\) are defined by \[P_0\mathbf x=\mathbf 0\] and \[P_m\mathbf x=x_1\mathbf e_1+\cdots+x_m\mathbf e_m \quad(1\leq m\leq n)\] Thus \(P_m\) is the projection whose range and null space are spanned by \(\{\mathbf e_1,\cdots,\mathbf e_m\}\) and \(\{\mathbf e_{m+1},\cdots,\mathbf e_n\}\), respectively.
Suppose \(\mathbf F\) is a \(\mathscr C'\)-mapping of an open set \(E\subset R^n\) into \(R^n\), \(\mathbf 0\in E,\quad \mathbf F(\mathbf 0)=\mathbf 0,\quad \mathbf F'(\mathbf 0)\) is invertible. Then there is a neighborhood of \(\mathbf 0\) in \(R^n\) in which a representation \[\mathbf F(\mathbf x)=\mathbf B_1\cdots \mathbf B_{n-1}\mathbf G_n\cdots \mathbf G_1(\mathbf x)\] is valid. Here, each \(\mathbf G_i\) is a primitive \(\mathscr C'\)-mapping in some neighborhood of \(\mathbf 0\); \(\mathbf G_i(\mathbf 0)=\mathbf 0, \quad\mathbf G_i'(\mathbf 0)\) is invertible, and each \(\mathbf B_i\) is either a flip or the identity operator. This means every invertible matrix can be decomposed into several Elementary Operations with the identity matrix.
Suppose \(T\) is a \(1-1\) \(\mathscr C'\)-mapping of an open set \(E\subset R^k\) into \(R^k\) such that the Jacobian of \(T\) \(J_T(\mathbf x)\ne 0\) for all \(\mathbf x\in E\), which implies that the inverse matrix of \(T\), \(T^{-1}\) is continuous on \(T(E)\). If \(f\) is a continuous function on \(R^k\) whose support is compact and lies in \(T(E)\), then \(f(T(\mathbf x))\) is compact \(\Biggl(\) Suppose \(f\) is a continuous mapping of a compact metric space \(X\) into a metric space \(Y\), then \(f(X)\) is compact. Let \(\{V_a\}\) be an open cover of \(f(X)\), since \(f\) is continuous so each of the sets \(f^{-1}(V_a)\) is open. Since \(X\) is compact, there are finitely many indices \(a_1,a_2,\cdots,a_n\), such that \[X\subset f^{-1}(V_{a_1})\cup\cdots\cup f^{-1}(V_{a_n})\] Since \(f(f^{-1}(E))\subset E\) for every \(E\subset Y\), then \[f(X)\subset V_{a_1}\cup \cdots\cup V_{a_n}\] \(\Biggr)\) and \[\int_{R^k}f(\mathbf y)d\mathbf y=\int_{R^k}f(T(\mathbf x))|J_T(\mathbf x)|d\mathbf x\] The point is that the integrals are integrals of functions over subsets of \(R^k\) and we associate no direction or orientation with these subsets.
To say that \(\mathbf f\) is a \(\mathscr C'\)-mapping (or a \(\mathscr C''\)-mapping) of a compact set \(D\subset R^k\) into \(R^n\) means that there is a \(\mathscr C'\)-mapping (or a \(\mathscr C''\)-mapping) \(\mathbf g\) of an open set \(W\subset R^k\) into \(R^n\) such that \(D\subset W\) and such that \(\mathbf g(\mathbf x)=\mathbf f(\mathbf x)\) for all \(\mathbf x\in D\).
Suppose \(E\) is an open set in \(R^n\), a k-surface in \(E\) is a \(\mathscr C'\)-mapping \(\Phi\) from a compact set \(D\subset R^k\) into \(E\). \(D\) is called the parameter domain of \(\Phi\). Points of \(D\) will be denoted by \(\mathbf u=(u_1,\cdots,u_k)\).
Suppose \(E\) is an open set in \(R^n\). A differential form of order \(k\ge 1\) in \(E\) (k-form in \(E\)) is a function \[\omega=\sum a_{i_1\cdots i_k}(\mathbf x)dx_{i_1}\land\cdots\land dx_{i_k}\] (where the indices \(i_1,\cdots,i_k\) range independently from \(1\) to \(n\),) which assigns to each k-surface \(\Phi\) in \(E\) a number \[\omega(\Phi)=\int_{\Phi}\omega=\int_D\sum a_{i_1\cdots i_k}(\Phi(\mathbf u))\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}d\mathbf u\] where \(D\) is the parameter domain of \(\Phi\) and the functions \(a_{i_1\cdots i_k}\) are assumed to be the real and continuous in \(E\). A k-form \(\omega\) is said to be of class \(\mathscr C'\) or \(\mathscr C''\) if the functions \(a_{i_1\cdots i_k}\) are all of class \(\mathscr C'\) or \(\mathscr C''\). The Jacobian \[\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}=\begin{vmatrix} \frac{\partial(x_{i_1})}{\partial(u_{1})}&\frac{\partial(x_{i_1})}{\partial(u_{2})}&\cdots&\frac{\partial(x_{i_1})}{\partial(u_{k})}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{\partial(x_{i_k})}{\partial(u_{1})}&\frac{\partial(x_{i_k})}{\partial(u_{2})}&\cdots&\frac{\partial(x_{i_k})}{\partial(u_{k})}\\ \end{vmatrix}\] will changes sign if two of its rows are interchanged, so that \[dx_i\land dx_j=-dx_j\land dx_i\] which is called anticommutative relation. \[dx_i\land dx_j=0\] if \(i=j\).
Spherical coordinates \((r,\theta,\phi)\) and rectangular coordinates \((x,y,z)\) are related as follows: \[x=r\sin\theta\cos\phi\] \[y=r\sin\theta\sin\phi\] \[z=r\cos\theta\] When \(D\) is the 3-cell defined by \((0\leq r\leq 1,\quad 0\leq\theta\leq\pi,\quad 0\leq\phi\leq 2\pi)\), define \(\Phi(r,\theta,\phi)=(x,y,z)\) Then \[J_{\Phi}(r,\theta,\phi)=\frac{\partial(x,y,z)}{\partial(r,\theta,\phi)}=\begin{vmatrix} \frac{\partial(x)}{\partial(r)}&\frac{\partial(x)}{\partial(\theta)}&\frac{\partial(x)}{\partial(\phi)}\\ \frac{\partial(y)}{\partial(r)}&\frac{\partial(y)}{\partial(\theta)}&\frac{\partial(y)}{\partial(\phi)}\\ \frac{\partial(z)}{\partial(r)}&\frac{\partial(z)}{\partial(\theta)}&\frac{\partial(z)}{\partial(\phi)}\\ \end{vmatrix}=\\ \begin{vmatrix} \sin\theta\cos\phi&r\cos\theta\cos\phi&-r\sin\theta\sin\phi\\ \sin\theta\sin\phi&r\cos\theta\sin\phi&r\sin\theta\cos\phi\\ \cos\theta&-r\sin\theta&0\\ \end{vmatrix}=r^2\sin\theta\] Hence \[\int_{\Phi}1\cdot dx\land dy\land dz=\int_{D}J_{\Phi}d\phi d\theta dr=\int_{0}^{1}\int_{0}^{\pi}\int_{0}^{2\pi}r^2\sin\theta d\phi d\theta dr=2\pi\int_{0}^{1}\int_{0}^{\pi}r^2\sin\theta d\theta dr=\frac{4\pi}{3}\] which is the volume of the sphere of radius \(1\).
Elementary properties Let \(\omega\),\(\omega_1\),\(\omega_2\) be k-forms in \(E\), we write \(\omega_1=\omega_2\) if and only if \(\omega_1(\Phi)=\omega_2(\Phi)\) for every k-surface \(\Phi\) in \(E\). \(\omega=0\) means \(\omega(\Phi)=0\) for every k-surface \(\Phi\) in \(E\). For a real number \(c\), \(c\omega\) is the k-form defined by \[\int_{\Phi}c\omega=c\int_{\Phi}\omega\] and \(\omega(\Phi)=\omega_1(\Phi)+\omega_2(\Phi)\) means that \[\int_{\Phi}\omega=\int_{\Phi}\omega_1+\int_{\Phi}\omega_2\] for every k-surface \(\Phi\) in \(E\). \(-\omega\) is defined so that \[\int_{\Phi}(-\omega)=-\int_{\Phi}\omega\]
If \(i_1,\cdots,i_k\) are integers such that \(1\leq i_1<i_2<\cdots<i_k\leq n\) and if \(I\) is the ordered k-tuple \(\{i_1,\cdots,i_k\}\), then we call \(I\) an increasing k-index and we use the notation \[dx_I=dx_{i_1}\land\cdots\land dx_{i_k}\] This forms \(dx_I\) are so-called basic k-forms in \(R^n\). There are precisely \[{n \choose k}=\frac{n!}{k!(n-k)!}\] basic k-forms in \(R^n\). Because each interchange of one pair in a k-form amounts to a multiplication by \(-1\), every k-form can be represented in terms of basic k-forms. If \((j_1,\cdots,j_n)\) is an ordered n-tuple of integers, define the sequence of the n-tuple \[s(j_1,\cdots,j_n)=\prod_{p<q}sgn(j_q-j_p)\] where \(\text{sgn }x=1\) if \(x>0\),\(\text{sgn }x=-1\) if \(x<0\),\(\text{sgn }x=0\) if \(x=0\). If every k-tuple in the k-form \[\omega=\sum a_{i_1\cdots i_k}(\mathbf x)dx_{i_1}\land\cdots\land dx_{i_k}\] is converted to an increasing k-index (multiplied by the sign of the sequence of the n-tuple), then we obtain the standard presentation of the k-form \[\omega=\sum_{I}b_I(\mathbf x)dx_I\]
If a standard presentation k-form in an open set \(E\subset R^n\) \[\omega=\sum_{I}b_I(\mathbf x)dx_I=0\] then \(b_I(\mathbf x)=0\) for every increasing k-index \(I\) and for every \(\mathbf x\in E\). Assume that \(b_J(\mathbf v)>0\) for some \(\mathbf v\in E\) and for some increasing k-index \(J=\{j_1,\cdots,j_k\}\), since \(b_J\) is continuous, there exists \(h>0\) such that \(b_J(\mathbf x)>0\) for all \(\mathbf x\in R^n\) whose coordinates satisfy \(|x_i-v_i|\leq h\). Let \(D\) be the k-cell in \(R^k\) such that \(\mathbf u\in D\) if and only if \(|u_r|\leq h\) for \(r=1,\cdots,k\). Define \[\Phi(\mathbf u)=\mathbf v+\sum_{r=1}^{k}u_r\mathbf e_{j_r}\quad(\mathbf u\in D)\] and \[\frac{\partial(x_{j_1},\cdots,x_{j_k})}{\partial(u_1,\cdots,u_k)}=\begin{vmatrix} \frac{\partial(x_{j_1})}{\partial(u_1)}&\cdots&\frac{\partial(x_{j_1})}{\partial(u_k)}\\ \vdots&\ddots&\vdots\\ \frac{\partial(x_{j_k})}{\partial(u_1)}&\cdots&\frac{\partial(x_{j_k})}{\partial(u_k)}\\ \end{vmatrix}=\begin{vmatrix} 1&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&1\\ \end{vmatrix}=1\] Thus \(\Phi\) is a k-surface in \(E\), with parameter domain \(D\), and \(b_J(\Phi(\mathbf u))>0\) for every \(\mathbf u\in D\) The k-surface \[\omega(\Phi)=\int_{\Phi}\omega=\int_D\sum a_{i_1\cdots i_k}(\Phi(\mathbf u))\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}d\mathbf u\\ =\int_D\sum a_{i_1\cdots i_k}(\Phi(\mathbf u))\cdot 1\cdot d\mathbf u=\int_{D}b_J(\Phi(\mathbf u))d\mathbf u>0\] since \(b_J(\Phi(\mathbf u))>0\). It follows that \(\omega(\Phi)\ne 0\), which gives a contradiction. So \(b_I(\mathbf x)=0\) for every increasing k-index \(I\) and for every \(\mathbf x\in E\).
Suppose \(I=\{i_1,\cdots,i_p\}\), \(J=\{j_1,\cdots,j_q\}\) where \(1\leq i_1<i_2<\cdots<i_p\leq n\) and \(1\leq j_1<j_2<\cdots<j_q\leq n\). The product of the basic forms: \(dx_I\land dx_J\) is a \((p+q)\)-form in \(R^n\), and \(dx_I\land dx_J=dx_{i_1}\land\cdots\land dx_{i_p}\land dx_{j_1}\land\cdots\land dx_{j_q}\) If \(I\) and \(J\) have elements in common, then \(dx_I\land dx_J=0\), if \(I\) and \(J\) have no elements in common, let us write \([I,J]\) for the increasing \((p+q)\)-index which is obtained by arranging the members of \(I\cup J\) in increasing order. Then \(dx_{[I,J]}\) is a basic \((p+q)\)-form and \[dx_I\land dx_J=(-1)^adx_{[I,J]}\] where \(a\) is the number of differences \(j_t-i_s\) that are negative.
Suppose \(\omega\) and \(\lambda\) are \(p-\) and \(q\)-forms in some open set \(E\subset R^n\) with standard presentations \[\omega=\sum_{I}b_I(\mathbf x)dx_I,\quad \lambda=\sum_{J}c_J(\mathbf x)dx_J\] where \(I\) and \(J\) range over all increasing \(p\)-indices and over all increasing \(q\)-indices taken from the set \(\{1,\cdots,n\}\). Their product is \[\omega\land\lambda=\sum_{I,J}b_I(\mathbf x)c_J(\mathbf x)dx_I\land dx_J\] Thus \(\omega\land\lambda\) is a \((p+q)\)-form in \(E\).
We define a differentiation operator \(d\) which associates a \((k+1)\)-form \(d\omega\) to each k-form \(\omega\) of class \(\mathscr C'\) in some open set \(E\subset R^n\). A \(0\)-form of class \(\mathscr C'\) in \(E\) is a real function \(f\in\mathscr C'(E)\), and \(df\) is \(1\)-form \[df=\sum_{i=1}^{n}(D_if)(\mathbf x)dx_i\] If \(\omega=\sum b_I(\mathbf x)dx_I\) is the standard presentation of a k-form \(\omega\), and \(b_I\in\mathscr C'(E)\) for each increasing k-index \(I\), then we define \[d\omega=\sum_{I}(db_I)\land dx_I\]
Suppose \(\omega\) and \(\lambda\) are \(p-\) and \(q\)-forms of class \(\mathscr C'\) in \(E\), then \[d(\omega\land\lambda)=(d\omega)\land\lambda+(-1)^p\omega\land d\lambda\] Because the new differential forms of \(d\lambda\) have to move \(p\) times to bypass the \(p\) differential forms of \(\omega\) to join the old differential forms of \(\lambda\). 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 \[d^2\omega=(d^2f)\land dx_I=0\]
change of variables of differential forms Suppose \(E\) is an open set \(R^n\), \(T\) is a \(\mathscr C'\)-mapping of \(E\) into an open set \(V\subset R^m\) and \(\omega\) is k-form in \(V\), whose standard presentation is \[\omega=\sum_{I}b_I(\mathbf y)dy_I\quad(\mathbf y\in V, \mathbf x\in E)\] Let \((t_1,\cdots,t_m)\) be the components of \(T\): If \[\mathbf y=(y_1,\cdots,y_m)=T(\mathbf x)\] Then \(y_i=t_i(\mathbf x)\) \[\underset{T:(m\times n)}{\begin{bmatrix} t_1\\ t_2\\ \vdots\\ t_m\\ \end{bmatrix}}\underset{(n\times 1)}{\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n\\ \end{bmatrix}}=\underset{(m\times 1)}{\begin{bmatrix} y_1\\ y_2\\ \vdots\\ y_m\\ \end{bmatrix}}\] \[dt_i=\sum_{j=1}^{n}(D_jt_i)(\mathbf x)dx_j\quad(1\leq i\leq m)\] Thus each \(dt_i\) is a 1-form in \(E\). The mapping \(T\) transforms a k-form \(\omega_T\) in \(E\) into \(\omega\) \[\omega_T=\sum_{I}b_I(T(\mathbf x))dt_{i_1}\land\cdots\land dt_{i_k}\] \(I=\{i_1,\cdots,i_k\}\) is an increasing k-index.
If \(\omega\) and \(\lambda\) are both k-forms in \(V\), then the standard presentations are \[\omega=\sum_{I}b_I(\mathbf y)dy_I\quad(\mathbf y\in V, \mathbf x\in E)\] \[\lambda=\sum_{I}a_I(\mathbf y)dy_I\quad(\mathbf y\in V, \mathbf x\in E)\] \[\omega+\lambda=\sum_{I}(b_I+a_I)(\mathbf y)dy_I\] then \[\begin{align} (\omega+\lambda)_T&=\sum_{I}(b_I+a_I)(T(\mathbf x))dt_{i_1}\land\cdots\land dt_{i_k}\\ &=\sum_{I}b_I(T(\mathbf x))dt_{i_1}\land\cdots\land dt_{i_k}+\sum_{I}a_I(T(\mathbf x))dt_{i_1}\land\cdots\land dt_{i_k}\\ &=\omega_T+\lambda_T \end{align}\]
If \(\omega\) and \(\lambda\) are k- and m-forms in \(V\), then the standard presentations are \[\omega=\sum_{I}b_I(\mathbf y)dy_I\quad(\mathbf y\in V, \mathbf x\in E)\] \[\lambda=\sum_{J}a_J(\mathbf y)dy_J\quad(\mathbf y\in V, \mathbf x\in E)\] The \((k+m)\)-form \[\omega\land\lambda=(\sum_{I}b_I(\mathbf y)dy_I)\land(\sum_{J}a_J(\mathbf y)dy_J)=\sum_{I,J}b_I(\mathbf y)a_J(\mathbf y)dy_I\land dy_J\] where \(I\) and \(J\) range independently over their possible values. If \(I\) and \(J\) have an element in common, \(dy_I\land dy_J=0\) If \(I\) and \(J\) have no element in common, \[\begin{align} (\omega\land\lambda)_T&=\Biggl(\sum_{I}b_I(T(\mathbf x))dt_{I}\Biggr)\land\Biggl(\sum_{J}a_J(T(\mathbf x))dt_{J}\Biggr)\\ &=\omega_T\land\lambda_T \end{align}\]
If \(f\) is a \(0\)-form of class \(\mathscr C'\) in \(V\), then \[f_T(\mathbf x)=f(T(\mathbf x)), \quad df=\sum_{i}(D_if)(\mathbf y)dy_i\] Write \(f_T(\mathbf x)=f(T(\mathbf x))\) using matrix \[d\underset{f:(1\times m)}{\begin{bmatrix} f \end{bmatrix}}\underset{T:(m\times n)}{\begin{bmatrix} t_1\\ t_2\\ \vdots\\ t_m\\ \end{bmatrix}}\underset{(n\times 1)}{\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n\\ \end{bmatrix}}=d\underset{f_T:(1\times n)}{\begin{bmatrix} f_T \end{bmatrix}}\underset{(n\times 1)}{\begin{bmatrix} x_1\\ x_2\\ \vdots\\ x_n\\ \end{bmatrix}}\] It means that \[\begin{align} d(f_T)&=\sum_{j=1}^{n}(D_jf_T(\mathbf x))dx_j\\ &=\sum_{j=1}^n\sum_{i=1}^m(D_if(T(\mathbf x)))(D_jt_i(\mathbf x))dx_j\\ &=\sum_{i=1}^m(D_if(T(\mathbf x)))dt_i\\ &=(df)_T\\ \end{align}\] If \(\omega=fdy_I\) then \[\omega_T=f_T(\mathbf x)(dy_I)_T\] and \[\begin{align} d(\omega_T)&=d\Biggl(f_T(\mathbf x)(dy_I)_T\Biggr)\\ &=d(f_T)\land(dy_I)_T+f_T\land d((dy_I)_T)\\ &=d(f_T)\land(dy_I)_T+f_T\land d(dt_I)\\ &=d(f_T)\land(dy_I)_T+f_T\land0\\ &=d(f_T)\land(dy_I)_T\\ &=(df)_T\land(dy_I)_T\\ &=\Bigl((df)\land(dy_I)\Bigr)_T\\ &=\Bigl(d\omega\Bigr)_T\quad(\omega=fdy_I)\\ \end{align}\]
Suppose \(T\) is a \(\mathscr C'\)-mapping of an open set \(E\subset R^n\) into an open set \(V\subset R^m\), \(S\) is a \(\mathscr C'\)-mapping of \(V\) into an open set \(W\subset R^p\), and \(\omega\) is k-form in \(W\), so that \(\omega_S\) is a k-form in \(V\) and both \((\omega_S)_T\) and \(\omega_{ST}\) are k-forms in \(E\) where \((ST)(\mathbf x)=S(T(\mathbf x))\). If both \(\omega\) and \(\lambda\) are forms in \(W\), then \[((\omega\land\lambda)_S)_T=(\omega_S\land\lambda_S)_T=(\omega_S)_T\land(\lambda_S)_T\] and \[(\omega\land\lambda)_{ST}=\omega_{ST}\land\lambda_{ST}\] Let \((t_1,\cdots,t_m)\) be the components of \(T\), let \((s_1,\cdots,s_p)\) be the components of \(S\), let \((r_1,\cdots,r_p)\) be the components of \(ST\) and denote the points of \(E,V,W\) by \(\mathbf x, \mathbf y, \mathbf z\) respectively. If \(\omega\) is \(0\)-form \[\omega=dz_q,\quad(q=1,\cdots,p)\] then \[\omega_S=ds_q=\sum_{j}^{}(D_js_q(\mathbf y))dy_j\] \[\begin{align} (\omega_S)_T&=(\sum_{j}^{}(D_js_q(\mathbf y))dy_j)_T\\ &=\sum_{j}^{}(D_js_q(T(\mathbf x)))dt_j\\ &=\sum_{j}^{}(D_js_q(T(\mathbf x)))\sum_{i}(D_it_j(\mathbf x))dx_i\\ &=\sum_{i}(D_ir_q(\mathbf x))dx_i\\ &=dr_q\\ &=\omega_{ST} \end{align}\]
Suppose \(\omega\) is a k-form in an open set \(E\subset R^n\), \(\Phi\) is a k-surface in \(E\), with parameter domain \(D\subset R^k\), and \(\Delta\) is the k-surface in \(R^k\), with parameter domain \(D\), defined by \(\Delta(\mathbf u)=\mathbf u\quad(\mathbf u\in D)\). If \[\omega=a(\mathbf x)dx_{i_1}\land\cdots\land dx_{i_k}\] then \[\omega_{\Phi}=a(\Phi(\mathbf u))d\phi_{i_1}\land\cdots\land d\phi_{i_k}\] where \((\phi_1,\cdots,\phi_n)\) are components of \(\Phi\). \[d\begin{bmatrix} \phi_{i_1}\\ \vdots\\ \phi_{i_p}\\ \vdots\\ \phi_{i_k}\\ \end{bmatrix}\mathbf u=\underset{(k\times k)(p=1,\cdots,k)}{\underbrace{\begin{bmatrix} \frac{\partial(\phi_{i_1})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_1})}{\partial(u_k)}\\ \vdots&\ddots&\vdots\\ \frac{\partial(\phi_{i_p})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_p})}{\partial(u_k)}\\ \vdots&\ddots&\vdots\\ \frac{\partial(\phi_{i_k})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_k})}{\partial(u_k)}\\ \end{bmatrix}}}\underset{(q=1,\cdots,k)}{\begin{bmatrix} du_1\\ \vdots\\ du_q\\ \vdots\\ du_k\\ \end{bmatrix}}\] Then \[d\phi_{i_p}=\sum_{q=1}^{k}\frac{\partial(\phi_{i_p})}{\partial(u_q)}du_q\] so that \[\begin{align} d\phi_{i_1}\land\cdots\land d\phi_{i_k}&=\Biggl(\sum_{q=1}^{k}\frac{\partial(\phi_{i_1})}{\partial(u_q)}du_q\Biggr)\land\cdots\land\Biggl(\sum_{q=1}^{k}\frac{\partial(\phi_{i_k})}{\partial(u_q)}du_q\Biggr)\\ &=\sum\frac{\partial(\phi_{i_1})}{\partial(u_{q_1})}\frac{\partial(\phi_{i_2})}{\partial(u_{q_2})}\cdots\frac{\partial(\phi_{i_k})}{\partial(u_{q_k})}du_{q_1}du_{q_2}\land\cdots\land du_{q_k}\\ &\Biggl[(q_1,\cdots,q_k)\text{ range independently over }(1,\cdots,k)\Biggr]\\ &=\sum\frac{\partial(\phi_{i_1})}{\partial(u_{q_1})}\frac{\partial(\phi_{i_2})}{\partial(u_{q_2})}\cdots\frac{\partial(\phi_{i_k})}{\partial(u_{q_k})}s(q_1,\cdots,q_k)du_{1}du_{2}\land\cdots\land du_{k}\\ &=det\begin{bmatrix} \frac{\partial(\phi_{i_1})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_1})}{\partial(u_k)}\\ \vdots&\ddots&\vdots\\ \frac{\partial(\phi_{i_p})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_p})}{\partial(u_k)}\\ \vdots&\ddots&\vdots\\ \frac{\partial(\phi_{i_k})}{\partial(u_1)}&\cdots&\frac{\partial(\phi_{i_k})}{\partial(u_k)}\\ \end{bmatrix}du_{1}du_{2}\land\cdots\land du_{k}\\ &=\frac{\partial(x_{i_1},\cdots,x_{i_k})}{\partial(u_1,\cdots,u_k)}du_{1}du_{2}\land\cdots\land du_{k}\\ &=J(\mathbf u)du_{1}du_{2}\land\cdots\land du_{k}\\ \end{align}\] then \[\omega_{\Phi}=a(\Phi(\mathbf u))d\phi_{i_1}\land\cdots\land d\phi_{i_k}=a(\Phi(\mathbf u))J(\mathbf u)du_{1}du_{2}\land\cdots\land du_{k}\] Then \[\begin{align} \int_{\Phi}\omega&=\int_{\Phi}a(\mathbf x)d\mathbf x\\ &=\int_{D}a(\Phi(\mathbf u))J(\mathbf u)d\mathbf u\\ &=\int_{\Delta}a(\Phi(\mathbf u))J(\mathbf u)du_{1}\land\cdots\land du_{k}\\ &=\int_{\Delta}\omega_{\Phi} \end{align}\]
Suppose \(T\) is a \(\mathscr C'\)-mapping of an open set \(E\subset R^n\) into an open set \(V\subset R^m\), \(\Phi\) is a k-surface in \(E\) and \(\omega\) is a k-form in \(V\), let \(D\) be the parameter domain of \(\Phi\) and \(\Delta\) is the k-surface in \(R^k\), with parameter domain \(D\), defined by \(\Delta(\mathbf u)=\mathbf u\quad(\mathbf u\in D)\). \[\underset{\Phi}{E}(\subset R^n)\xrightarrow{T(\in \mathscr C')}\underset{\omega}{V}(\subset R^m)\] Then \[\begin{align} \int_{T\Phi}\omega&=\int_{\Delta}\omega_{T\Phi}\\ &=\int_{\Delta}(\omega_T)_\Phi\\ &=\int_{\Phi}\omega_T\\ \end{align}\]