# 数学代写|偏微分方程代写partial difference equations代考|The lemma of Poincar´e

#### Doug I. Jones

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## 数学代写|偏微分方程代写partial difference equations代考|The lemma of Poincar´e

The theory of curvilinear integrals was transferred to the higher-dimensional situation of surface-integrals especially by de Rham (compare G. de Rham: Varietés differentiables, Hermann, Paris 1955). In this context we refer the reader to Paragraph 20 in the textbook by $\mathrm{H}$. Holmann and H. Rummler: $A l$ ternierende Differentialformen, BI-Wissenschaftsverlag, 2.Auflage, 1981.

We shall construct primitives for arbitrary $m$-forms, which correspond to vector-potentials – however, in ‘contractible domains’ only. Here we do not need the Stokes integral theorem!

Definition 1. A continuous $m$-form with $1 \leq m \leq n$ in an open set $\Omega \subset \mathbb{R}^n$ with $n \in \mathbb{N}$, namely
$$\omega=\sum_{1 \leq i_1<\ldots<i_m \leq n} a_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m}, \quad x \in \Omega$$
is named exact if we have an $(m-1)$-form
$$\lambda=\sum_{1 \leq i_1<\ldots<i_{m-1} \leq n} b_{i_1 \ldots i_{m-1}}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}}, \quad x \in \Omega$$
of the class $C^1(\Omega)$ with the property
$$d \lambda=\omega \quad \text { in } \Omega .$$
We begin with the easy
Theorem 1. An exact differential form $\omega \in C^1(\Omega)$ is closed.
Proof: We calculate
\begin{aligned} d \omega & =d(d \lambda)=d \sum_{1 \leq i_1<\ldots<i_{m-1} \leq n} d b_{i_1 \ldots i_{m-1}}(x) \wedge d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}} \ & =\sum_{1 \leq i_1<\ldots<i_{m-1} \leq n}\left(d d b_{i_1 \ldots i_{m-1}}(x)\right) \wedge d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}}=0, \end{aligned}
which implies the statement above.
q.e.d.
We now provide a condition on the domain $\Omega$, which guarantees that a closed differential form is necessarily exact.

## 数学代写|偏微分方程代写partial difference equations代考|Co-derivatives and the Laplace-Beltrami operator

In this section we introduce an inner product for differential forms. We consider the space
$$\mathbb{R}^n:=\left{\bar{x}=\left(\bar{x}1, \ldots, \bar{x}_n\right): \bar{x}_i \in \mathbb{R}, i=1, \ldots, n\right}$$ with the subset $\Theta \subset \mathbb{R}^n$. Furthermore, we have given two continuous $m$-forms on $\Theta$, namely $$\bar{\alpha}:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\bar{x}) d \bar{x}{i_1} \wedge \ldots \wedge d \bar{x}{i_m}, \quad \bar{x} \in \Theta,$$ as well as $$\bar{\beta}:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{b}{i_1 \ldots i_m}(\bar{x}) d \bar{x}{i_1} \wedge \ldots \wedge d \bar{x}{i_m}, \quad \bar{x} \in \Theta$$ We define an inner product between the $m$-forms $\bar{\alpha}$ and $\bar{\beta}$ as follows: $$(\bar{\alpha}, \bar{\beta})_m:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\bar{x}) \bar{b}{i_1 \ldots i_m}(\bar{x}), \quad m=0,1, \ldots, n .$$
Consequently, the inner product attributes a 0 -form to a pair of $m$-forms. It represents a symmetric bilinear form on the vector space of $m$-forms.
Now we consider the parameter transformation
$$\bar{x}=\Phi(x)=\left(\Phi_1\left(x_1, \ldots, x_n\right), \ldots, \Phi_n\left(x_1, \ldots, x_n\right)\right): \Omega \longrightarrow \Theta \in C^2(\Omega)$$

on the open set $\Omega \subset \mathbb{R}^n$. The mapping $\Phi$ satisfies
$$J_{\Phi}(x)=\operatorname{det}(\partial \Phi(x)) \neq 0 \quad \text { for all } \quad x \in \Omega .$$
We set
$$g(x):=\left(J_{\Phi}(x)\right)^2=\operatorname{det}\left(\partial \Phi(x)^t \circ \partial \Phi(x)\right), \quad x \in \Omega .$$
The volume form
$$\omega=\sqrt{g(x)} d x_1 \wedge \ldots \wedge d x_n, \quad x \in \Omega$$
is associated with the transformation $\bar{x}=\Phi(x)$ in a natural way. The $m$-forms $\bar{\alpha}$ and $\bar{\beta}$ are transformed into the $m$-forms
\begin{aligned} \alpha:=\bar{\alpha}{\Phi} & =\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\Phi(x)) d \Phi{i_1}(x) \wedge \ldots \wedge d \Phi_{i_m}(x) \ & =: \sum_{1 \leq i_1<\ldots<i_m \leq n} a_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m} \end{aligned}
and
\begin{aligned} \beta:=\bar{\beta}{\Phi} & =\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{b}{i_1 \ldots i_m}(\Phi(x)) d \Phi{i_1}(x) \wedge \ldots \wedge d \Phi_{i_m}(x) \ & =: \sum_{1 \leq i_1<\ldots<i_m \leq n} b_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m}, \end{aligned}
respectively. We shall define an inner product $(\alpha, \beta)_m$ between the transformed $m$-forms $\alpha$ and $\beta$ such that it is parameter-invariant:
$$(\alpha, \beta)_m(x)=(\bar{\alpha}, \bar{\beta})_m(\Phi(x)), \quad x \in \Omega$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The lemma of Poincar´e

$$\omega=\sum_{1 \leq i_1<\ldots<i_m \leq n} a_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m}, \quad x \in \Omega$$

$$\lambda=\sum_{1 \leq i_1<\ldots<i_{m-1} \leq n} b_{i_1 \ldots i_{m-1}}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}}, \quad x \in \Omega$$

$$d \lambda=\omega \quad \text { in } \Omega .$$

\begin{aligned} d \omega & =d(d \lambda)=d \sum_{1 \leq i_1<\ldots<i_{m-1} \leq n} d b_{i_1 \ldots i_{m-1}}(x) \wedge d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}} \ & =\sum_{1 \leq i_1<\ldots<i_{m-1} \leq n}\left(d d b_{i_1 \ldots i_{m-1}}(x)\right) \wedge d x_{i_1} \wedge \ldots \wedge d x_{i_{m-1}}=0, \end{aligned}

Q.E.D.

## 数学代写|偏微分方程代写partial difference equations代考|Co-derivatives and the Laplace-Beltrami operator

$$\mathbb{R}^n:=\left{\bar{x}=\left(\bar{x}1, \ldots, \bar{x}_n\right): \bar{x}_i \in \mathbb{R}, i=1, \ldots, n\right}$$和子集$\Theta \subset \mathbb{R}^n$。此外，我们在$\Theta$上给出了两个连续的$m$ -表单，即$$\bar{\alpha}:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\bar{x}) d \bar{x}{i_1} \wedge \ldots \wedge d \bar{x}{i_m}, \quad \bar{x} \in \Theta,$$和$$\bar{\beta}:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{b}{i_1 \ldots i_m}(\bar{x}) d \bar{x}{i_1} \wedge \ldots \wedge d \bar{x}{i_m}, \quad \bar{x} \in \Theta$$。我们定义了$m$ -表单$\bar{\alpha}$和$\bar{\beta}$之间的内积如下:$$(\bar{\alpha}, \bar{\beta})_m:=\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\bar{x}) \bar{b}{i_1 \ldots i_m}(\bar{x}), \quad m=0,1, \ldots, n .$$

$$\bar{x}=\Phi(x)=\left(\Phi_1\left(x_1, \ldots, x_n\right), \ldots, \Phi_n\left(x_1, \ldots, x_n\right)\right): \Omega \longrightarrow \Theta \in C^2(\Omega)$$

$$J_{\Phi}(x)=\operatorname{det}(\partial \Phi(x)) \neq 0 \quad \text { for all } \quad x \in \Omega .$$

$$g(x):=\left(J_{\Phi}(x)\right)^2=\operatorname{det}\left(\partial \Phi(x)^t \circ \partial \Phi(x)\right), \quad x \in \Omega .$$

$$\omega=\sqrt{g(x)} d x_1 \wedge \ldots \wedge d x_n, \quad x \in \Omega$$

\begin{aligned} \alpha:=\bar{\alpha}{\Phi} & =\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{a}{i_1 \ldots i_m}(\Phi(x)) d \Phi{i_1}(x) \wedge \ldots \wedge d \Phi_{i_m}(x) \ & =: \sum_{1 \leq i_1<\ldots<i_m \leq n} a_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m} \end{aligned}

\begin{aligned} \beta:=\bar{\beta}{\Phi} & =\sum{1 \leq i_1<\ldots<i_m \leq n} \bar{b}{i_1 \ldots i_m}(\Phi(x)) d \Phi{i_1}(x) \wedge \ldots \wedge d \Phi_{i_m}(x) \ & =: \sum_{1 \leq i_1<\ldots<i_m \leq n} b_{i_1 \ldots i_m}(x) d x_{i_1} \wedge \ldots \wedge d x_{i_m}, \end{aligned}

$$(\alpha, \beta)_m(x)=(\bar{\alpha}, \bar{\beta})_m(\Phi(x)), \quad x \in \Omega$$

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