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# 数学代写|偏微分方程代考Partial Differential Equations代写|The degree of mapping in R

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## 数学代写|偏微分方程代考Partial Differential Equations代写|The degree of mapping in R

J.L.E. Brouwer introduced the degree of mapping in $\mathbb{R}^n$ by simplicial approximation within combinatorial topology. When we intend to define the degree of mapping analytically, we have to replace the integral of the winding number by $(n-1)$-dimensional surface-integrals in $\mathbb{R}^n$ (compare G. de Rham: Varietés differentiables). E. Heinz transformed the boundary integral for the winding number into an area-integral and thus created a possibility to define the degree of mapping in $\mathbb{R}^n$ in a natural way. We present the transition of the winding-number-integral to the area-integral in $\mathbb{R}^2$ in the sequel:

Let the radius $R \in(0,+\infty)$ and the function $f=f(z) \in C^2\left(B_R, \mathbb{C}\right)$ satisfying $\varphi(t):=f\left(R e^{i t}\right) \neq 0,0 \leq t \leq 2 \pi$ be given. We choose $\varepsilon>0$ so small that $\varepsilon<|\varphi(t)|$ for all $t \in[0,2 \pi]$ holds true. Now we consider a function
$$\psi(r)=\left{\begin{array}{l} 0,0 \leq r \leq \delta \ 1, \varepsilon \leq r \end{array} \in C^1([0,+\infty), \mathbb{R})\right.$$

with $0<\delta<\varepsilon$, and we investigate the winding-number-integral
\begin{aligned} 2 \pi i W(\varphi) & =\oint_{\partial B_R}\left{\frac{f_x}{f} d x+\frac{f_y}{f} d y\right}=\oint_{\partial B_R} d F \ & =\oint_{\partial B_R} \psi(|f(z)|) d F(z)=\oint_{\partial B_R} \psi(|f(x, y)|) d F(x, y) \end{aligned}
with
$$F(x, y)=\log f(x, y)+2 \pi i k, \quad k \in \mathbb{Z} .$$
We remark that $F$ is defined only locally; however, the differential $d F$ is globally available. The 1 -form
$$\psi(|f(x, y)|) d F(x, y), \quad(x, y) \in B_R$$
belongs to the class $C^1\left(B_R\right)$, and we determine its exterior derivative. Via the identity
\begin{aligned} d{\psi(|f(x, y)|)} & =\psi^{\prime}(|f(x, y)|)\left{\left((f \cdot \bar{f})^{\frac{1}{2}}\right)_x d x+\left((f \cdot \bar{f})^{\frac{1}{2}}\right)_y d y\right} \ & =\frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{f\left(\bar{f}_x d x+\bar{f}_y d y\right)+\bar{f}\left(f_x d x+f_y d y\right)\right} \end{aligned}
we obtain
\begin{aligned} d{\psi(|f|) d F}= & d{\psi(|f|)} \wedge d F \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{f\left(\bar{f}_x d x+\bar{f}_y d y\right)+\bar{f}\left(f_x d x+f_y d y\right)\right} \ & \wedge\left{\frac{1}{f}\left(f_x d x+f_y d y\right)\right} \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{\bar{f}_x d x+\bar{f}_y d y\right} \wedge\left{f_x d x+f_y d y\right} \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{\left(\bar{f}_x d x \wedge f_y d y\right)-\left(\overline{f_x d x \wedge f_y d y}\right)\right} \ = & i \frac{\psi^{\prime}(|f(x, y)|)}{|f(x, y)|} \operatorname{Im}\left{\bar{f}_x d x \wedge f_y d y\right} \end{aligned}

## 数学代写|偏微分方程代考Partial Differential Equations代写|Geometric existence theorems

We begin with the fundamental
Proposition 1. Let $\Omega \subset \mathbb{R}^n$ denote a bounded open set and define the function $f(x)=\varepsilon(x-\xi), x \in \Omega$; here we choose $\varepsilon= \pm 1$ and $\xi \in \Omega$. Then we have the identity $d(f, \Omega)=\varepsilon^n$.

Proof: We take a number $\eta>0$ such that $|f(x)|>\eta$ for all points $x \in \partial \Omega$ holds true. Let $\omega \in C_0^0((0, \eta), \mathbb{R})$ denote an arbitrary test function satisfying
$$\int_{\mathbb{R}^n} \omega(|x|) d x=1$$

Then we have
$$d(f, \Omega)=\int_{\Omega} \omega(|f(x)|) J_f(x) d x=\int_{\Omega} \omega(|x-\xi|) \varepsilon^n d x=\varepsilon^n .$$
q.e.d.
Theorem 1. Let $f_\tau(x)=f(x, \tau): \bar{\Omega} \times[a, b] \rightarrow \mathbb{R}^n \in C^0\left(\bar{\Omega} \times[a, b], \mathbb{R}^n\right)$ denote a family of mappings with
$$f_\tau(x) \neq 0 \quad \text { for all } \quad x \in \partial \Omega \quad \text { and all } \quad \tau \in[a, b] .$$
Furthermore, we have the function
$$f_a(x)=(x-\xi), \quad x \in \Omega$$
with a point $\xi \in \Omega$. For each parameter $\tau \in[a, b]$ we find a point $x_\tau \in \Omega$ satisfying $f\left(x_\tau, \tau\right)=0$.
Proof: The homotopy lemma and Proposition 1 yield
$$d\left(f_\tau, \Omega\right)=d\left(f_a, \Omega\right)=1 \quad \text { for all } \tau \in[a, b] .$$
Consequently, there exists a point $x_\tau \in \Omega$ with $f\left(x_\tau, \tau\right)=0$ for each parameter $\tau \in[a, b]$, due to Theorem 3 in $\S 2$.
q.e.d.

# 偏微分方程代写

## 数学代写|偏微分方程代考Partial Differential Equations代写|The degree of mapping in R

J.L.E. browwer在组合拓扑中引入了$\mathbb{R}^n$中简单逼近的映射度。当我们打算解析地定义映射的程度时，我们必须用$\mathbb{R}^n$中的$(n-1)$维曲面积分来代替圈数的积分(比较G. de Rham:可变的可变的)。E. Heinz将圈数的边界积分转化为面积积分，从而创造了在$\mathbb{R}^n$中以自然的方式定义映射度的可能性。在续文中，我们给出了$\mathbb{R}^2$中圈数积分到面积积分的转换:

$$\psi(r)=\left{\begin{array}{l} 0,0 \leq r \leq \delta \ 1, \varepsilon \leq r \end{array} \in C^1([0,+\infty), \mathbb{R})\right.$$

\begin{aligned} 2 \pi i W(\varphi) & =\oint_{\partial B_R}\left{\frac{f_x}{f} d x+\frac{f_y}{f} d y\right}=\oint_{\partial B_R} d F \ & =\oint_{\partial B_R} \psi(|f(z)|) d F(z)=\oint_{\partial B_R} \psi(|f(x, y)|) d F(x, y) \end{aligned}

$$F(x, y)=\log f(x, y)+2 \pi i k, \quad k \in \mathbb{Z} .$$

$$\psi(|f(x, y)|) d F(x, y), \quad(x, y) \in B_R$$

\begin{aligned} d{\psi(|f(x, y)|)} & =\psi^{\prime}(|f(x, y)|)\left{\left((f \cdot \bar{f})^{\frac{1}{2}}\right)_x d x+\left((f \cdot \bar{f})^{\frac{1}{2}}\right)_y d y\right} \ & =\frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{f\left(\bar{f}_x d x+\bar{f}_y d y\right)+\bar{f}\left(f_x d x+f_y d y\right)\right} \end{aligned}

\begin{aligned} d{\psi(|f|) d F}= & d{\psi(|f|)} \wedge d F \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{f\left(\bar{f}_x d x+\bar{f}_y d y\right)+\bar{f}\left(f_x d x+f_y d y\right)\right} \ & \wedge\left{\frac{1}{f}\left(f_x d x+f_y d y\right)\right} \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{\bar{f}_x d x+\bar{f}_y d y\right} \wedge\left{f_x d x+f_y d y\right} \ = & \frac{\psi^{\prime}(|f(x, y)|)}{2|f(x, y)|}\left{\left(\bar{f}_x d x \wedge f_y d y\right)-\left(\overline{f_x d x \wedge f_y d y}\right)\right} \ = & i \frac{\psi^{\prime}(|f(x, y)|)}{|f(x, y)|} \operatorname{Im}\left{\bar{f}_x d x \wedge f_y d y\right} \end{aligned}

## 数学代写|偏微分方程代考Partial Differential Equations代写|Geometric existence theorems

$$\int_{\mathbb{R}^n} \omega(|x|) d x=1$$

$$d(f, \Omega)=\int_{\Omega} \omega(|f(x)|) J_f(x) d x=\int_{\Omega} \omega(|x-\xi|) \varepsilon^n d x=\varepsilon^n .$$
Q.E.D.

$$f_\tau(x) \neq 0 \quad \text { for all } \quad x \in \partial \Omega \quad \text { and all } \quad \tau \in[a, b] .$$

$$f_a(x)=(x-\xi), \quad x \in \Omega$$

$$d\left(f_\tau, \Omega\right)=d\left(f_a, \Omega\right)=1 \quad \text { for all } \tau \in[a, b] .$$

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