# 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|MATH263 Tangent Spaces

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Tangent Spaces

We have used, informally, the following proposition: If $S$ is a manifold in $\mathbb{R}^{n}$, and $(x, f(x))$ is tangent to $S$ for each $x \in S$, then $S$ is an invariant manifold for the differential equation $\dot{x}=f(x)$. To make this proposition precise, we will give a definition of the concept of a tangent vector on a manifold. This definition is the main topic of this section.

Let us begin by considering some examples where the proposition on tangents and invariant manifolds can be applied.

The vector field on $\mathbb{R}^{3}$ associated with the system of differential equations given by
\begin{aligned} &\dot{x}=x(y+z), \ &\dot{y}=-y^{2}+x \cos z, \ &\dot{z}=2 x+z-\sin y \end{aligned}
is “tangent” to the linear two-dimensional submanifold $S:={(x, y, z)$ : $x=0}$ in the following sense: If $(a, b, c) \in S$, then the value of the vector function
$$(x, y, z) \mapsto\left(x(y+z), y^{2}+x \cos z, 2 x+z-\sin y\right)$$
at $(a, b, c)$ is a vector in the linear space $S$. Note that the vector assigned by the vector field depends on the point in $S$. For this reason, we will view the vector field as the function
$$(x, y, z) \mapsto\left(x, y, z, x(y+z),-y^{2}+x \cos z, 2 x+z-\sin y\right)$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代考|Change of Coordinates

The proof of Proposition $1.70$ contains an important computation that is useful in many other contexts; namely, the formula for changing coordinates in an autonomous differential equation. To reiterate this result, suppose that we have a differential equation $\dot{x}=f(x)$ where $x \in \mathbb{R}^{n}$, and $S \subseteq \mathbb{R}^{n}$ is an invariant $k$-dimensional submanifold. If $g$ is a diffeomorphism from $S$ to some $k$-dimensional submanifold $M \subseteq \mathbb{R}^{m}$, then the ordinary differential equation (or, more precisely, the vector field associated with the differential equation) can be “pushed forward” to $M$. In fact, if $g: S \rightarrow M$ is the diffeomorphism, then
$$\dot{y}=D g\left(g^{-1}(y)\right) f\left(g^{-1}(y)\right)$$
is a differential equation on $M$. Since $g$ is a diffeomorphism, the new differential equation is the same as the original one up to a change of coordinates as schematically depicted in Figure 1.16.

Example 1.75. Consider $\dot{x}=x-x^{2}, x \in \mathbb{R}$. Let $S={x \in \mathbb{R}: x>0}$, $M=S$, and let $g: S \rightarrow M$ denote the diffeomorphism defined by $g(x)=$ $1 / x$. Here, $g^{-1}(y)=1 / y$ and
\begin{aligned} \dot{y} &=\operatorname{Dg}\left(g^{-1}(y)\right) f\left(g^{-1}(y)\right) \ &=-\left(\frac{1}{y}\right)^{-2}\left(\frac{1}{y}-\frac{1}{y^{2}}\right) \ &=-y+1 \end{aligned}

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考|Tangent Spaces

$$\dot{x}=x(y+z), \quad \dot{y}=-y^{2}+x \cos z, \dot{z}=2 x+z-\sin y$$

$$(x, y, z) \mapsto\left(x(y+z), y^{2}+x \cos z, 2 x+z-\sin y\right)$$

$$(x, y, z) \mapsto\left(x, y, z, x(y+z),-y^{2}+x \cos z, 2 x+z-\sin y\right)$$

## 微积分网课代修|常微分方程代写Ordinary Differential Equation代 考|Change of Coordinates

$M \subseteq \mathbb{R}^{m}$ ，则常微分方程 (或更准确地说，与微分方程相关的矢量场) 可以“向前推” 为 $M$. 事实上，如果 $g: S \rightarrow M$ 是微分同胚，那么
$$\dot{y}=D g\left(g^{-1}(y)\right) f\left(g^{-1}(y)\right)$$

$$\dot{y}=\operatorname{Dg}\left(g^{-1}(y)\right) f\left(g^{-1}(y)\right) \quad=-\left(\frac{1}{y}\right)^{-2}\left(\frac{1}{y}-\frac{1}{y^{2}}\right)=-y+1$$