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# 数学代写|拓扑学代写TOPOLOGY代考|The Fundamental Theorem of Algebra

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## 数学代写|拓扑学代写TOPOLOGY代考|The Fundamental Theorem of Algebra

Complex Numbers. We begin with a quick review of some facts about complex numbers. Complex numbers are numbers of the form $z=x+i y$, where $x, y \in \mathbb{R}$ and $i$ satisfies $i^2=-1$. In other words, $i$ and also $-i$ are roots of the polynomial $p(z)=z^2+1$

• We can add two complex numbers: $\left(x_1+i y_1\right)+\left(x_2+i y_2\right)=\left(x_1+x_2\right)+i\left(y_1+\right.$ $\left.y_2\right)$.
• We can multiply a complex number by a real number: $c(x+i y)=c x+i c y$.
Thus, in this respect, complex numbers behave just like points in $\mathbb{R}^2$ – the complex number $x+i y$ becomes the point $(x, y)$ and now addition and real number multiplication of complex numbers become vector addition and scalar multiplication in $\mathbb{R}^2$
• We can also multiply one complex number by another. The product $\left(x_1+i y_1\right)\left(x_2+\right.$ $i y_2$ ) is found by fully multiplying these two brackets out, and replacing $i^2$ by -1 when it occurs. The answer is $\left(x_1 x_2-y_1 y_2\right)+i\left(y_1 x_2+y_2 x_1\right)$.

In $\mathbb{R}^2$, we can use polar coordinates to represent points. We represent $(x, y) \in \mathbb{R}^2$ by its distance from the origin $r=\sqrt{x^2+y^2}$, and the angle $\theta$ made by the line connecting $(x, y)$ to $(0,0)$, so that $\tan (\theta)=y / x$. Now $(x, y)=(r \cos (\theta), r \sin (\theta))$. We can thus also use polar coordinates to describe complex numbers.

• The length of a complex number $z=x+i y$ is denoted $|z|=\sqrt{x^2+y^2}$.
• The polar angle of $z$ is denoted $\arg (z)$, and $\tan (\arg (z))=y / x$.
• Now $z=|z|(\cos (\arg (z))+i \sin (\arg (z)))$.
• De Moivre’s Theorem states that $(\cos (\theta)+i \sin (\theta))^n=\cos (n \theta)+i \sin (n \theta)$. This formula can be proven easily by induction on $n$, using the angle-sum formulae for cosine and sine. It follows that $z^n=|z|^n(\cos (n \arg (z))+i \sin (n \arg (z)))$.
• We therefore define $e^{i \theta}=\cos (\theta)+i \sin (\theta)$, because it has the analogous property $\left(e^{i \theta}\right)^n=e^{i n \theta}$. (There’s more to this, but we won’t get into it here.)
• Note that $\left|e^{i \theta}\right|=1$ and that as $\theta \in[0,2 \pi]$ advances, the curve $\alpha(\theta)=e^{i \theta}$ traces out the unit circle in $\mathbb{C}$ when it is viewed as $\mathbb{R}^2$.
• As a consequence, the curve $\gamma(\theta)=e^{i n \theta}$ traces out $n$ windings of the unit circle for $\theta \in[0,2 \pi]$.

## 数学代写|拓扑学代写TOPOLOGY代考|Further Applications of the Fundamental Group

The Borsuk-Ulam Theorem. The Borsuk-Ulam Theorem is a classical result in topology that asserts the existence of a special kind of point (the solution of an equation) based on very minimal assumptions!

Theorem 11.3 (Borsuk-Ulam) Suppose $f: \mathbb{S}^2 \rightarrow \mathbb{R}^2$ is a continuous function from the sphere to the plane. Then there exists $x \in \mathbb{S}^2$ so that $f(x)=f(-x)$.

Therefore, no matter the function, there exist two antipodal points on the sphere with identical function values. A surprising, silly application of this theorem (which everyone is contractually obligated to mention when first discussing the BorsukUlam Theorem) is that there exists a pair of antipodal points on the surface of the Earth (or a perfectly spherical version of the Earth, at least, so that we can define antipodes) where the temperature and atmospheric pressure are exactly the same.
Proof Suppose the Borsuk-Ulam Theorem is false. Define a new function $\widehat{f}: \mathbb{S}^2 \rightarrow$ $\mathbb{R}^2$ by $\widehat{f}(x):=f(x)-f(-x)$, which by our assumption on the falsehood of the Borsuk-Ulam Theorem is actually a function $\widehat{f}: \mathbb{S}^2 \rightarrow \mathbb{R}^2 \backslash{(0,0)}$. Note that $\widehat{f}$ is an odd function because $\widehat{f}(-x)=-\widehat{f}(x)$. Next, let $\alpha:[0,2 \pi] \rightarrow \mathbb{S}^2$ be the curve that winds once around the equator, i.e. $\alpha(s):=(\cos (s), \sin (s), 0)$. The curve $\widehat{f} \circ \alpha$ : $[0,2 \pi] \rightarrow \mathbb{R}^2 \backslash{(0,0)}$ now winds a certain number of times around the origin $(0,0)$; this number is $\operatorname{deg}_{(0,0)}(\widehat{f} \circ \alpha)$. This number has to be zero because we can easily construct a homotopy of the curve $\alpha$ to a point by sliding it upwards to the north pole of $\mathbb{S}^2$. Hence this homotopy must also allow the curve $\widehat{f} \circ \alpha$ to shrink continuously to a point inside $\mathbb{R}^2 \backslash{(0,0)}$. Consequently, the degree of the curve $\widehat{f} \circ \alpha$, as defined in Chapter 10, is zero.

We can now reach a contradiction, because we can actually show that the degree of $\widehat{f} \circ \alpha$ has to be an odd number. This is due to the following calculation:
\begin{aligned} \widehat{f} \circ \alpha(s+\pi) & =\widehat{f}(\cos (s+\pi), \sin (s+\pi), 0) \ & =\widehat{f}(-\cos (s),-\sin (s), 0) \ & =-\widehat{f}(\cos (s), \sin (s), 0) \ & =-\widehat{f} \circ \alpha(s) . \end{aligned}

## 数学代写|拓扑学代写TOPOLOGY代考|The Fundamental Theorem of Algebra

• 我们可以添加两个复数:
$$\left(x_1+i y_1\right)+\left(x_2+i y_2\right)=\left(x_1+x_2\right)+i\left(y_1+y_2\right) .$$
• 我们可以将一个复数乘以一个实数: $c(x+i y)=c x+i c y$.
因此，在这方面，复数的行为就像中的点 $\mathbb{R}^2-$ 复数 $x+i y$ 成为重点 $(x, y)$ 现在 复数的加法和实数乘法变成了向量加法和标量乘法 $\mathbb{R}^2$
• 我们还可以将一个复数乘以另一个。产品 $\left(x_1+i y_1\right)\left(x_2+i y_2\right)$ 是通过将这两 个括号完全相乘并替换 $i^2$ 发生时减 -1 。答案是
$$\left(x_1 x_2-y_1 y_2\right)+i\left(y_1 x_2+y_2 x_1\right) \text {. }$$
在 $\mathbb{R}^2$ ，我们可以用极坐标来表示点。我们代表 $(x, y) \in \mathbb{R}^2$ 通过它到原点的距离 $r=\sqrt{x^2+y^2}$ ，和角度 $\theta$ 由连接线制成 $(x, y)$ 到 $(0,0)$ ，以便 $\tan (\theta)=y / x$. 现在 $(x, y)=(r \cos (\theta), r \sin (\theta))$. 因此，我们也可以使用极坐标来描述复数。
• 复数的长度 $z=x+i y$ 表示为 $|z|=\sqrt{x^2+y^2}$.
• 的极角 $z$ 表示为 $\arg (z)$ ， 和 $\tan (\arg (z))=y / x$.
• 现在 $z=|z|(\cos (\arg (z))+i \sin (\arg (z)))$.
• De Moivre 定理指出 $(\cos (\theta)+i \sin (\theta))^n=\cos (n \theta)+i \sin (n \theta)$. 这个公 式可以很容易地通过归纳证明 $n$ ，使用余弦和正弦的角和公式。它遵循 $z^n=|z|^n(\cos (n \arg (z))+i \sin (n \arg (z)))$.
• 因此我们定义 $e^{i \theta}=\cos (\theta)+i \sin (\theta)$ ，因为它具有类似的属性 $\left(e^{i \theta}\right)^n=e^{i n \theta}$. (还有更多内容，但我们不会在这里讨论。)
• 注意 $\left|e^{i \theta}\right|=1$ 那作为 $\theta \in[0,2 \pi]$ 进步，曲线 $\alpha(\theta)=e^{i \theta}$ 追踪出单位圆 $C$ 当它 被视为 $\mathbb{R}^2$.
• 因此，曲线 $\gamma(\theta)=e^{i n \theta}$ 查出 $n$ 单位圆的绕组为 $\theta \in[0,2 \pi]$.

## 数学代写|拓扑学代写TOPOLOGY代考|Further Applications of the Fundamental Group

Borsuk-Ulam 定理。Borsuk-Ulam 定理是拓扑学中的经典结果，它基于非常小的假 设断言存在一种特殊的点 (方程的解) !

$$\widehat{f} \circ \alpha(s+\pi)=\widehat{f}(\cos (s+\pi), \sin (s+\pi), 0) \quad=\widehat{f}(-\cos (s),-\sin (s), 0)=-\widehat{f}(\cos (s), \sin (s), 0)$$

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