Type $\theta$ Stokes' Theorem for Chains

TL;DR

This paper introduces a generalized version of Stokes' theorem, called type θ Stokes' theorem, which extends the fundamental theorem of calculus to higher-dimensional chains.

Contribution

It proposes a new theoretical framework for type θ k-chains, extending classical calculus theorems to higher dimensions.

Findings

Established the type θ Stokes' theorem for k-chains.

Extended fundamental theorem of calculus to higher-dimensional chains.

Provided a new mathematical foundation for differential chains.

Abstract

We present type $\theta$ Stokes' theorem for type $\theta$ $k$-chains which extends the fundamental theorem of calculus in higher dimensions.

Equations

$$(x_1+x_2{1}^{\#})(y_1+y_2{1}^{\#})=x_1{y}_1+(x_1{y}_2+x_2{y}_1)1^{\#},$$

$$\mbox{either}\quad\left\{\begin{array}[]{c}Re\,x>Re\,y\\ Ze\,x\geq Ze\,y\end{array}\right.\quad\mbox{or}\quad\left\{\begin{array}[]{c}Re\,x=Re\,y\\ Ze\,x>Ze\,y\end{array}\right.$$

$$\mbox{either}\quad\left\{\begin{array}[]{c}Re\,x>Re\,y\\ Ze\,x\leq Ze\,y\end{array}\right.\quad\mbox{or}\quad\left\{\begin{array}[]{c}Re\,x=Re\,y\\ Ze\,x<Ze\,y\end{array}\right.$$

$${\mathcal{R}}^{(2)n}:=\{(x^1,x^2,\dots ,x^n)|x^i\in {\mathcal{R}}^{(2)},i=1,2,\dots ,n\}.$$

$$x+y:=(x^1+y^1,x^2+y^2,\dots ,x^n+y^n),ax:=(a{x}^1,a{x}^2,\dots ,a{x}^n),$$

$$||x||:=\Big{(}2\displaystyle\sum_{i=1}^{n}(x^{i1})^{2}+\displaystyle\sum_{i=1}^{n}(x^{i2})^{2}\Big{)}^{\frac{1}{2}}.$$

$$N(a;ϵ)=\{x\in {\mathcal{R}}^{(2)n}|||x-a||<ϵ\}\text{mbox}and{N}^{\ast }(a;ϵ):=N(a;ϵ)\setminus \{a\}.$$

$$\underset{x\to a}{\lim }\frac{||f(x)-f(a)-\lambda (x-a)||}{||x-a||}=0,$$

$$ϵ>0$$

$$\delta >0$$

$$x\in N^{\ast }(a;\delta )\Rightarrow \frac{||f(x)-f(a)-\lambda (x-a)||}{||x-a||}<ϵ.$$

$$x\in N(a;\delta )\cap D\Rightarrow ||f(x)-f(a)||<ϵ.$$

$$\mathcal{D}(g\circ f)(a)=\mathcal{D}g\big{(}f(a)\big{)}\circ\mathcal{D}f(a).$$

$$a^{i}=x^{i(0)}\stackrel{{\scriptstyle\theta}}{{<}}x^{i(1)}\stackrel{{\scriptstyle\theta}}{{<}}\dots\stackrel{{\scriptstyle\theta}}{{<}}x^{i(N_{i})}=b^{i}\quad\mbox{for $1\leq i\leq n$.}$$

$$S=S_{j_1,\dots ,j_n}=[x^{1(j_1-1)},x^{1(j_1)}{]}_{\theta }\times [x^{2(j_2-1)},x^{2(j_2)}{]}_{\theta }\times \cdots \times [x^{n(j_n-1)},x^{n(j_n)}{]}_{\theta },$$

$$v(S)=v(S_{j_{1},\dots,j_{n}})=\Big{(}x^{1(j_{1})}-x^{1(j_{1}-1)}\Big{)}\cdots\Big{(}x^{n(j_{n})}-x^{n(j_{n}-1)}\Big{)}.$$

$$f(x)=f_{Re}(x)+f_{Ze}(x)\,1^{\#}\quad\mbox{for $x\in A$},$$

$$f_{\clubsuit}(x)=f_{\clubsuit}(x^{11},x^{12},x^{21},x^{22},\dots,x^{n1},x^{n2})\quad\mbox{with $\clubsuit\in\{Re,\,Ze\}$}$$

$$\sup_{S}f_{\clubsuit}:=\sup\big{\{}f_{\clubsuit}(x)\,|\,x\in S\big{\}}$$

$$\inf_{S}f_{\clubsuit}:=\inf\big{\{}f_{\clubsuit}(x)\,|\,x\in S\big{\}}$$

$$m\leq f_{\clubsuit}(x)\leq M\quad\mbox{for all $x\in A$ and $\clubsuit\in\{Re,\,Ze\}$}.$$

$$\underset{S}{\sup }{f}_{♣}\le M\text{mbox}and\underset{S}{\inf }{f}_{♣}\ge m$$

$$U_{\theta }(P,f)$$

$$L_{\theta }(P,f)$$

$$\displaystyle\clubsuit\Big{(}\big{(}m+1^{\#}m\big{)}\,v(A)\Big{)}\stackrel{{\scriptstyle 1}}{{\leq}}\clubsuit L_{1}(P,f)$$

$$\displaystyle\qquad\stackrel{{\scriptstyle 1}}{{\leq}}\clubsuit U_{1}(P,f)\stackrel{{\scriptstyle 1}}{{\leq}}\clubsuit\Big{(}\big{(}M+1^{\#}M\big{)}\,v(A)\Big{)}$$

$$\displaystyle Re\Big{(}\big{(}m+1^{\#}M\big{)}\,v(A)\Big{)}\stackrel{{\scriptstyle 2}}{{\leq}}Re\,L_{2}(P,f)$$

$$\displaystyle\qquad\stackrel{{\scriptstyle 2}}{{\leq}}Re\,U_{2}(P,f)\stackrel{{\scriptstyle 2}}{{\leq}}Re\Big{(}\big{(}M+1^{\#}m\big{)}\,v(A)\Big{)}$$

$$\displaystyle Ze\Big{(}\big{(}m+1^{\#}M\big{)}\,v(A)\Big{)}\stackrel{{\scriptstyle 2}}{{\geq}}Ze\,L_{2}(P,f)$$

$$\displaystyle\qquad\stackrel{{\scriptstyle 2}}{{\geq}}Ze\,U_{2}(P,f)\stackrel{{\scriptstyle 2}}{{\geq}}Ze\Big{(}\big{(}M+1^{\#}m\big{)}\,v(A)\Big{)}.$$

$$\mathcal{P}_{\theta}:=\Big{\{}P\,\Big{|}\,\mbox{$P$ is a partition of $[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$}\Big{\}}.$$

$$\big{\{}\clubsuit U_{\theta}(P,f)\,\big{|}\,P\in\mathcal{P}_{\theta}\big{\}},\quad\big{\{}\clubsuit L_{\theta}(P,f)\,\big{|}\,P\in\mathcal{P}_{\theta}\big{\}}\quad\mbox{with}\quad\clubsuit\in\{Re,\,Ze\}$$

$$\underline{{\int }_A}f(x)d_{\theta }V=\underline{{\int }_A}f(x^1,\dots ,x^n)d{x}^1\cdots d{x}^n$$

Full text

The fundamental concepts and the important theorems in [1] have natural extentions if the real number field $\mathcal{R}$ is replaced by the dual real number algebra $\mathcal{R}^{(2)}$ and the real linear transformations are replaced by $\mathcal{R}^{(2)}$-module maps. In this paper, we present type $\theta$ Stokes’ theorem for type $\theta$ $k$-chains which extends the ordinary Stokes’ theorem for chains.

Section 1 gives the basic definitions about dual real $n$-space. Section 2 introduces dual real differentiation for dual real-valued functions with dual real multivariable by using $\mathcal{R}^{(2)}$-module maps. Section 3 initiates the study of two types of integrations based on the two order relations on the dual real number algebra $\mathcal{R}^{(2)}$. Section 4 and Section 5 outline the notations and fundamental properties of alternating dual real tensors and dual real differential forms. Section 6, which is the last section of this paper, defines the type $\theta$ integral of a dual real differentiable $k$-form and presents type $\theta$ Stokes’ theorem for type $\theta$ $k$-chains.

1 Dual Real $n$-Space

Let $\mathcal{R}$ be the real number field with the identity $1$. Th dual real number algebra is the 2-dimensional real associative algebra $\mathcal{R}^{(2)}=\mathcal{R}\oplus\mathcal{R}\,1^{\#}$ with the following product

[TABLE]

where $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}$ are real numbers, and $\{1,\,1^{\#}\}$ is a basis of the 2-dimensional real vector space $\mathcal{R}^{(2)}$. If $x=x_{1}+x_{2}\,1^{\#}\in\mathcal{R}^{(2)}$ with $x_{1}$, $x_{2}\in\mathcal{R}$, then $Re\,x:=x_{1}$ and $Ze\,x:=x_{2}$ are called the real part and the zero-divisor part of $x$, respectively. The dual real number algebra $\mathcal{R}^{(2)}$ is a commutative real associative algebras with zero-divisors.

There are two generalized order relations on $\mathcal{R}^{(2)}$ which are compatible with the multiplication in $\mathcal{R}^{(2)}$.

Definition 1.1

Let $x$ and $y$ be two elements of $\mathcal{R}^{(2)}$.

(i)

We say that $x$ is type 1 greater than $y$ ( or $y$ is type 1 less than $x$) and we write $x\stackrel{{\scriptstyle 1}}{{>}}y$ (or $y\stackrel{{\scriptstyle 1}}{{<}}x$) if

[TABLE]

(ii)

We say that $x$ is type 2 greater than $y$ ( or $y$ is type 2 less than $x$) and we write $x\stackrel{{\scriptstyle 2}}{{>}}y$ (or $y\stackrel{{\scriptstyle 2}}{{<}}x$) if

[TABLE]

In the following of this paper, the letter $\theta$ always means an element in the set $\{1,2\}$. We use $x\stackrel{{\scriptstyle\theta}}{{\geq}}y$ when $x\stackrel{{\scriptstyle\theta}}{{>}}y$ or $x=y$.

For any positive integer $n$, the dual real $n$-space is the set

[TABLE]

An element $x=(x^{1},x^{2},\dots,x^{n})$ of $\mathcal{R}^{(2)n}$ is called a dual real vector or a point, and the dual real number $x^{i}=x^{i1}+x^{i2}\,1^{\#}$ with $x^{i1}$, $x^{i2}\in\mathcal{R}$ is called the $i$-th component of $x$ for $1\leq i\leq n$. the dual real $n$-space $\mathcal{R}^{(2)n}$ is a left $\mathcal{R}^{(2)}$-module with respect to the following addition and scalar multiplication:

[TABLE]

where $x=(x^{1},x^{2},\dots,x^{n})$, $y=(y^{1},y^{2},\dots,y^{n})\in\mathcal{R}^{(2)n}$ and $a\in\mathcal{R}^{(2)}$.

Let $S$ be a non-empty subset of left $\mathcal{R}^{(2)}$ -module $V$. An element of the left $\mathcal{R}^{(2)}$ -module $V$ is also called a dual real vector. $S$ is called $\mathcal{R}^{(2)}$-linearly dependent if there exist a finite number of distinct dual real vectors $v_{1}$, $v_{2}$, $\dots$, $v_{n}$ in $S$ and deal real numbers $c_{1}$, $c_{2}$, $\dots$, $c_{n}$, not all zero, such that $c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}=0$. $S$ is called $\mathcal{R}^{(2)}$-linearly independent if $S$ is not $\mathcal{R}^{(2)}$-linearly dependent. $S$ is called spanning set of $v$ if every dual real vector in $V$ is a $\mathcal{R}^{(2)}$-linearly combination of dual real vectors of $S$, i.e., for every $v\in V$, there exist a finite number of distinct dual real vectors $v_{1}$, $v_{2}$, $\dots$, $v_{n}$ in $S$ and deal real numbers $c_{1}$, $c_{2}$, $\dots$, $c_{n}$ such that $v=c_{1}v_{1}+c_{2}v_{2}+\dots+c_{n}v_{n}$. $S$ is called a $\mathcal{R}^{(2)}$-basis for the left $\mathcal{R}^{(2)}$ -module $V$ if $S$ is a $\mathcal{R}^{(2)}$-linearly independent spanning set of $V$.

If $V$ is a free left $\mathcal{R}^{(2)}$-module having a finite $\mathcal{R}^{(2)}$-basis, then every $\mathcal{R}^{(2)}$-basis for $V$ contains the same number of dual real vectors. The unique number of dual real vectors in each $\mathcal{R}^{(2)}$-basis for $V$ is called the $\mathcal{R}^{(2)}$-dimension of $V$ and is denoted by $\dim_{\mathcal{R}^{(2)}}(V)$.

The dual real $n$-space $\mathcal{R}^{(2)n}$ is a free $\mathcal{R}^{(2)}$-module. Let $e_{1}:=(1,0,0,\dots,0)$, $e_{2}:=(0,1,0,\dots,0)$, $\dots$, $e_{n}:=(0,0,\dots,0,1)$. Then $\{e_{1},e_{2},\dots,e_{n}\}$ is a $\mathcal{R}^{(2)}$-basis for $\mathcal{R}^{(2)n}$ and is called the standard $\mathcal{R}^{(2)}$-basis.

2 Dual Real Differentiation

For $x=\displaystyle\sum_{i=1}^{n}(x^{i1}+1^{\#}x^{i2})e_{i}$ with $x^{i1}$, $x^{i2}\in\mathcal{R}$ for $1\leq i\leq n$, we define

[TABLE]

Then the real-valued function $||\,\,||:\mathcal{R}^{(2)n}\to\mathcal{R}$ defined by (1) is a norm on the dual real $n$-space $\mathcal{R}^{(2)n}$.

The dual real $n$-space $\mathcal{R}^{(2)n}$ is a metric space with the distance function $||\,\,||$ defined by (1). If $a\in\mathcal{R}^{(2)n}$ and $\epsilon\in\mathcal{R}$, we use $N(a;\,\epsilon)$ and $N^{*}(a;\,\epsilon)$ to denote the ordinary $\epsilon$-neighborhood and deleted $\epsilon$-neighborhood of $a$, respectively, i.e.,

[TABLE]

Definition 2.1

Let $U$ be an open subset of $\mathcal{R}^{(2)n}$. A function $f:U\to\mathcal{R}^{(2)m}$ is dual real differentiable at $a\in U$ if there is a $\mathcal{R}^{(2)}$-module map $\lambda:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)m}$ such that

[TABLE]

i.e., the following property holds:

[TABLE]

The $\mathcal{R}^{(2)}$-module map $\lambda$ is denoted by $\mathcal{D}f(a)$ and called the dual real derivative of $f$ at $a$. If $f$ is dual real differentiable at each point of the open subset $U$, then $f$ is said to be dual real differentiable on $U$.

Definition 2.2

Let $D$ be a subset of $\mathcal{R}^{(2)n}$. A function $f:D\to\mathcal{R}^{(2)m}$ is said to be continuous at $a\in D$ if for each positive real number $\epsilon>0$, there exists a positive real number $\delta>0$ such that

[TABLE]

If $f$ is continuous at each point of the open subset $D$, then $f$ is said to be continuous on $D$.

One can check that if $f:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)m}$ is dual real differentiable at $a\in\mathcal{R}^{(2)n}$, then $f$ is continuous at $a$. Also, the composition of two dual real differentiable functions is dual real differentiable. In fact, we have

Proposition 2.1

(Chain Rule)* If $f:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)m}$ is dual real differentiable at $a\in\mathcal{R}^{(2)n}$, and $g:\mathcal{R}^{(2)m}\to\mathcal{R}^{(2)p}$ is dual real differentiable at $f(a)\in\mathcal{R}^{(2)m}$, then the composition $g\circ f:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)p}$ is dual real differentiable at $a$, and*

[TABLE]

3 Type $\theta$ Integrations

Let $a^{i}$, $b^{i}\in\mathcal{R}^{(2)}$ with $a^{i}\stackrel{{\scriptstyle\theta}}{{<}}b^{i}$ for $1\leq i\leq n$. Recall that the type $\theta$ closed interval $[a^{i},b^{i}]_{\theta}$ is defined by [a^{i},b^{i}]_{\theta}:=\big{\{}\,x\in\mathcal{R}^{(2)}\,|\,a^{i}\stackrel{{\scriptstyle\theta}}{{\leq}}x\stackrel{{\scriptstyle\theta}}{{\leq}}b^{i}\,\big{\}}. The subset $[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$ of $\mathcal{R}^{(2)n}$ is called a type $\theta$ closed rectangle. A partition of a type $\theta$ closed rectangle $[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$ is a collection $P=(P_{1},\dots,P_{n})$, where $P_{i}$ is a partition of $[a^{i},b^{i}]_{\theta}$ with the partition points:

[TABLE]

The partition $P=(P_{1},\dots,P_{n})$ produces $N_{1}N_{2}\cdots N_{n}$ type $\theta$ closed subrectangles:

[TABLE]

where $1\leq j_{1}\leq N_{1}$, $\dots$ , $1\leq j_{n}\leq N_{n}$, which are called the type $\theta$ subrectangles of the partition $P$. The dual real volume $v(S)=v(S_{j_{1},\dots,j_{n}})$ is defined by

[TABLE]

Clearly, $v(S)=v(S_{j_{1},\dots,j_{n}})\stackrel{{\scriptstyle\theta}}{{\geq}}0$.

If $f:A\to\mathcal{R}^{(2)}$ is a function on a type $\theta$ closed rectangle $A$ in $\mathcal{R}^{(2)n}$, then

[TABLE]

where $x=(x^{1},\dots,x^{n})=(x^{11}+x^{12}\,1^{\#},\dots,x^{n1}+x^{n2}\,1^{\#})$ and

[TABLE]

is a real-valued function of $2n$ real variables $x^{11}$, $x^{12}$, $\dots$, $x^{n1}$, $x^{n2}$. We say that the function $f:A\to\mathcal{R}^{(2)}$ given by (2) is bounded if both $f_{Re}$ and $f_{Ze}$ are bounded on $A$ ($\subseteq\mathcal{R}^{(2)n}$).

Suppose that $A$ is a type $\theta$ closed rectangle in $\mathcal{R}^{(2)n}$, $f:A\to\mathcal{R}^{(2)n}$ is a bounded function, and $P=(P_{1},\dots,P_{n})$ is a partition of $A$. For each type $\theta$ subrectangle $S$ of the partition $P$, let f\big{|}S:S\to\mathcal{R}^{(2)} be the restriction of $f$ to $S$. The range of f\big{|}S is a bounded subset of the real number field $\mathcal{R}$. Hence, both

[TABLE]

and

[TABLE]

exist for $\clubsuit\in\{Re,\,Ze\}$ and each type $\theta$ subrectangle $S$ of the partition $P$. Since $f_{\clubsuit}$ is bounded on $A$, there exist real numbers $m$, $M\in\mathcal{R}$ such that

[TABLE]

Clearly, we have

[TABLE]

for $\clubsuit\in\{Re,\,Ze\}$ and any type $\theta$ subrectangle $S$ of the partition $P$.

We define the type $\theta$ upper sum $U_{\theta}(P,f)$ of $f$ with respect to the partition $P$ of the type $\theta$ rectangle $A$ to be

[TABLE]

and the type $\theta$ lower sum $L_{\theta}(P,f)$ of $f$ with respect to the partition $P$ of the type $\theta$ rectangle $A$ to be

[TABLE]

where the notation $S\,\widehat{\in}\,P$ means that $S$ is a type $\theta$ subrectangle $S$ of the partition $P$.

Using (7) and (11), we have

[TABLE]

for $\clubsuit\in\{Re,\,Ze\}$ and

[TABLE]

and

[TABLE]

Let $\mathcal{P}_{\theta}$ be the set of all partitions of the type $\theta$ closed rectangle $A=[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$, i.e.,

[TABLE]

By (3), (3) and (3), the following four sets

[TABLE]

are bounded subsets of the real number field $\mathcal{R}$. Hence, the supremums and infimums of the four sets exist. We now define the type $\theta$ lower integral

[TABLE]

and the type $\theta$ upper integral

[TABLE]

of $f(x)$ on the type $\theta$ closed rectangle $A=[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$ by

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

If the type $\theta$ lower integral and the type $\theta$ upper integral of $f(x)$ on $A=[a^{1},b^{1}]_{\theta}\times\cdots\times[a^{n},b^{n}]_{\theta}$ are equal, i.e., if

[TABLE]

then we say that $f$ is type $\theta$ integrable on $A$, we denote their common value by $\displaystyle\int_{A}f(x)d_{\theta}V$ which is called the type $\theta$ integral of $f$ on $A$.

A basic fact is that if $f:A\to\mathcal{R}^{(2)}$ is a continuous function on a type $\theta$ closed rectangle $A$ in $\mathcal{R}^{(2)n}$, then $f$ is type $\theta$ integrable on $A$.

4 Alternating Dual Real Tensors

Let $V$ be a left $\mathcal{R}^{(2)}$-module, and let $k$ be a positive integer. We define $V^{k}$ by $V^{k}:=\underbrace{V\times\cdots\times V}_{k}$. A function $T:V^{k}\to\mathcal{R}^{(2)}$ is called a dual real $k$-tensor if $T$ is $\mathcal{R}^{(2)}$-multilinear, i.e., if for each $i$ with $1\leq i\leq k$ we have

[TABLE]

and

[TABLE]

where $v_{i}$, $v_{i}^{\prime}\in V$ and $a\in\mathcal{R}^{(2)}$. Let

[TABLE]

Then $\mathbf{T}^{k}(V)$ becomes a left $\mathcal{R}^{(2)}$-module if for $S$, $T\in\mathbf{T}^{k}(V)$ and $a\in\mathcal{R}^{(2)}$ we define

[TABLE]

The tensor product $S\otimes T\in\mathbf{T}^{k+\ell}(V)$ of $S\in\mathbf{T}^{k}(V)$ and $T\in\mathbf{T}^{\ell}(V)$ is defined by

[TABLE]

A dual real $k$-tensor $w\in\mathbf{T}^{k}(V)$ is called alternating if

[TABLE]

The set of all alternating dual real $k$-tensors is a left $\mathcal{R}^{(2)}$-submodule of $\mathbf{T}^{k}(V)$, which is denoted by $\mathbf{\Lambda}^{k}(V)$.

If $T\in\mathbf{T}^{k}(V)$, we define $Alt(T)$ by

[TABLE]

where $\mathbf{S}_{k}$ is the set of all permutations on the set $\{1,2,\dots,k\}$, and $sgn\,\sigma$ is the sign of a permutation $\sigma\in\mathbf{S}_{k}$.

For $w\in\mathbf{\Lambda}^{k}(V)$ and $\eta\in\mathbf{\Lambda}^{\ell}(V)$, the wedge product $w\wedge\eta\in\mathbf{\Lambda}^{k+\ell}(V)$ is defined by

[TABLE]

We finish this section with the following

Proposition 4.1

Let $v_{1}$, $\dots$, $v_{n}$ be a $\mathcal{R}^{(2)}$-basis for a left $\mathcal{R}^{(2)}$-module $V$. If $\varphi_{1}$, $\dots$, $\varphi_{n}$ is the dual $\mathcal{R}^{(2)}$-basis of $V$ with respect to the $\{v_{1},\dots,v_{n}\}$, i.e., $\varphi_{i}:V\to\mathcal{R}^{(2)}$ is a $\mathcal{R}^{(2)}$-module map satisfying $\varphi_{i}(v_{j}):=\delta_{ij}$ for $1\leq i,\,j\leq n$, then the set of all wedge products

[TABLE]

is a $\mathcal{R}^{(2)}$-basis of $\mathbf{\Lambda}^{k}(V)$. Therefore, \dim_{\mathcal{R}^{(2)}}\big{(}\mathbf{\Lambda}^{k}(V)\big{)}=\left(\begin{array}[]{c}n\\ k\end{array}\right)=\displaystyle\frac{n!}{k!(n-k)!}.

5 Dual Real Differential Forms

For $p\in\mathcal{R}^{(2)n}$, let $\mathcal{R}^{(2)n}_{p}$ be the set defined by $\mathcal{R}^{(2)n}_{p}:=\{\,(p,v)\,|\,v\in\mathcal{R}^{(2)n}\}$. The pair $(p,v)$ is also denoted by $v_{p}$. $\mathcal{R}^{(2)n}_{p}$ is made into a left $\mathcal{R}^{(2)}$-module by defining

[TABLE]

for $v$, $w\in\mathcal{R}^{(2)n}$ and $a\in\mathcal{R}^{(2)}$. Let $\{e_{1},e_{2},\dots,e_{n}\}$ be the standard $\mathcal{R}^{(2)}$-basis for $\mathcal{R}^{(2)n}$. Then $\{(e_{1})_{p},(e_{2})_{p},\dots,(e_{n})_{p}\}$ is a $\mathcal{R}^{(2)}$-basis for $\mathcal{R}^{(2)n}_{p}$, which is called the standard $\mathcal{R}^{(2)}$-basis for $\mathcal{R}^{(2)n}_{p}$.

A function $w:\mathcal{R}^{(2)n}\to\displaystyle\bigcup_{q\in\mathcal{R}^{(2)n}}\mathbf{\Lambda}^{k}(\mathcal{R}^{(2)n}_{q})$ is called a dual real $k$-differential form if there exist dual real differentiable functions $w_{i_{1},\dots,i_{k}}:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)}$ for $1\leq i_{1}<\dots<i_{k}\leq n$ such that

[TABLE]

for all $p\in\mathcal{R}^{(2)n}$, where $\{\varphi_{i_{1}}(p),\dots,\varphi_{i_{n}}(p)\}$ is the dual $\mathcal{R}^{(2)}$-basis with respect to the standard $\mathcal{R}^{(2)}$-basis $\{(e_{1})_{p},(e_{2})_{p},\dots,(e_{n})_{p}\}$ of $\mathcal{R}^{(2)n}_{p}$.

If $f:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)}$ is dual real differentiable, then $\mathcal{D}f(p)\in\mathbf{\Lambda}^{1}(\mathcal{R}^{(2)n})$ for all $p\in\mathcal{R}^{(2)n}$, and we can define a dual real $1$-form $df$ by

[TABLE]

By (16), we know that $(df)(p)\in\mathbf{\Lambda}^{1}(\mathcal{R}^{(2)n}_{p})$ for $p\in\mathcal{R}^{(2)n}$.

Since the $i$th-projection function $\pi^{i}:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)}$ is $\mathcal{R}^{(2)}$-linear, we get

[TABLE]

After denoting the dual real differential $1$-form $d\pi^{i}$ by $dx^{i}$, we have

[TABLE]

where $v=(v^{1},\dots,v^{n})$ and $v^{i}\in\mathcal{R}^{(2)}$ for $1\leq i\leq n$. It follows from (17) that $(dx^{i})(p)=\varphi_{i}(p)$, i.e., $\{dx^{1}(p),\dots,dx^{n}(p)\}$ is just the dual real $\mathcal{R}^{(2)}$-basis of the standard $\mathcal{R}^{(2)}$-basis $\{(e_{1})_{p},(e_{2})_{p},\dots,(e_{n})_{p}\}$ for $\mathcal{R}^{(2)n}_{p}$. Hence, every dual real differential $k$-form $w$ given by (15) can be written as

[TABLE]

The dual real differential $dw$ of $w$ is a dual real differentiable $(k+1)$-form on $\mathcal{R}^{(2)n}$, where $dw$ is defined by

[TABLE]

If $f:\mathcal{R}^{(2)n}\to\mathcal{R}^{(2)m}$ is dual real differentiable, we get a $\mathcal{R}^{(2)}$-linear map $f^{*}:\mathbf{\Lambda}^{k}(\mathcal{R}^{(2)m}_{f(p)})\to\mathbf{\Lambda}^{k}(\mathcal{R}^{(2)n}_{p})$, where $f^{*}$ is given by

[TABLE]

where $T\in\mathbf{\Lambda}^{k}(\mathcal{R}^{(2)m}_{f(p)})$ and $v_{1}$, $\dots$, $v_{k}\in\mathcal{R}^{(2)n}$. We can therefore define a dual real differentiable $k$-form on $\mathcal{R}^{(2)n}$ by

[TABLE]

which means that if $v_{1}$, $\dots$, $v_{k}\in\mathcal{R}^{(2)n}$, then $(f^{*}w)(p)\in\mathbf{\Lambda}^{k}(\mathcal{R}^{(2)n}_{p})$ and

[TABLE]

6 Type $\theta$ Stokes’ Theorem for Type $\theta$ $k$-Chain

For a non-negative integer $k$, we define $[0,\,b]_{\theta}^{0}:=\{0\}$ and

[TABLE]

where $\theta\in\{1,2\}$, $[0,\,b]_{\theta}\subseteq\mathcal{R}^{(2)}$ is a type $\theta$ closed interval, $b=1+(-1)^{\theta-1}r1^{\#}$ and $r\geq 0$ is a nonnegative real number. A type $\theta$ singular $k$-cube in $A\subseteq\mathcal{R}^{(2)n}$ is a function $c^{b,\theta}:[0,\,b]_{\theta}^{k}\to A$ which extends to be a dual real differentiable function on some open set $U$ of $\mathcal{R}^{(2)k}$ such that $[0,\,b]_{\theta}^{k}\subseteq U$. A finite sum of type $\theta$ singular $k$-cubes in $A$ with integer coefficients is called a type $\theta$ $k$-chain in $A$. A type $\theta$ singular [math]-cube in $A$ is a point in $A\subseteq\mathcal{R}^{(2)n}$. The type $\theta$ standard singular $k$-cube is the function $I^{k,b,\theta}:[0,\,b]_{\theta}^{k}\to\mathcal{R}^{(2)k}$ defined by

[TABLE]

For $k\geq 1$, $j\in\{1,\dots,k\}$ and $\alpha\in\{0,\,1\}$, we define a type $\theta$ singular $(k-1)$-cube $I^{k,b,\theta}_{(j,\alpha)}:[0,\,b]_{\theta}^{k-1}\to[0,\,b]_{\theta}^{k}\subseteq\mathcal{R}^{(2)k}$ by

[TABLE]

where $x=(x^{1},\dots,x^{k-1})\in[0,\,b]_{\theta}^{k-1}$ and $x^{1}$, $\dots$, $x^{k-1}\in[0,\,b]_{\theta}$. $I^{k,b,\theta}_{(j,0)}$ and $I^{k,b,\theta}_{(j,1)}$ are called the $(j,0)$-face of $I^{k,b,\theta}$ and $(j,1)$-face of $I^{k,b,\theta}$, respectively.

The boundary $\partial I^{k,b,\theta}$ of $I^{k,b,\theta}$ is the type $\theta$ $(k-1)$-chain defined by

[TABLE]

For a general type $\theta$ singular $k$-cube $c^{b,\theta}:[0,\,b]_{\theta}^{k}\to A\subseteq\mathcal{R}^{(2)n}$, the $(j,\alpha)$-face $c^{b,\theta}_{j,\alpha}$ of $c^{b,\theta}$ is defined by

[TABLE]

and the boundary $\partial c^{b,\theta}$ of $c^{b,\theta}$ is defined by

[TABLE]

If $a_{i}$ is an integer and $c^{b,\theta}_{i}$ is a type $\theta$ singular $k$-cube for $1\leq i\leq m$, then the boundary of type $\theta$ $k$-chain $\displaystyle\sum_{i=1}^{m}a_{i}\,c^{b,\theta}_{i}$ is defined by

[TABLE]

Let $w$ be a dual real differentiable $k$-form on $[0,\,b]_{\theta}^{k}\subseteq\mathcal{R}^{(2)k}$. Then there exists a unique dual real differentiable function $f(x^{1},\dots,x^{k})$ such that

[TABLE]

Using the uniqueness of the function $f(x^{1},\dots,x^{k})$ and the type $\theta$ integral introduced in section 3, we define

[TABLE]

which is also written as

[TABLE]

If $w$ is a dual real differentiable $k$-form on $A\subseteq\mathcal{R}^{(2)n}$ and $c^{b,\theta}:[0,\,b]_{\theta}^{k}\to A$ is a type $\theta$ singular $k$-cube in $A$ with $k\geq 1$, we define

[TABLE]

where $(c^{b,\theta})^{*}\,w$ is the dual real differentiable $k$-form on $[0,\,b]_{\theta}^{k}$ defined by (18).

If $w$ is a dual real differentiable [math]-form and $c^{b,\theta}:[0,\,b]_{\theta}^{0}=\{0\}\to A$ is (type $\theta$) singular [math]-cube in $A$, we define

[TABLE]

The type $\theta$ integral of a dual real differentiable $k$-form over a type $\theta$ $k$-chain $c^{b,\theta}=\displaystyle\sum_{i=1}^{m}a_{i}\,c^{b,\theta}_{i}$ is defined by

[TABLE]

Proposition 6.1

(Type $\theta$ Stokes’ Theorem for Type $\theta$ $k$-Chain) If $w$ is a dual real differentiable $(k-1)$-form on an open set $A\subseteq\mathcal{R}^{(2)n}$ and $c^{b,\theta}$ is a type $\theta$ $k$-chain in $A$ with $k\geq 1$, then

[TABLE]