# 数学代写|实分析作业代写Real analysis代考|MATH450

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## 数学代写|实分析作业代写Real analysis代考|Integral Curves

If $U$ is an open subset of $\mathbb{R}^n$, then a vector field on $U$ may be defined as a function $X: U \rightarrow \mathbb{R}^n$. The vector field is smooth if $X$ is a smooth function. In classical notation, $X$ is written $X=\sum_{j=1}^n a_j\left(x_1, \ldots, x_n\right) \frac{\partial}{\partial x_j}$, and the function carries $\left(x_1, \ldots, x_n\right)$ to $\left(a_1\left(x_1, \ldots, x_n\right), \ldots, a_n\left(x_1, \ldots, x_n\right)\right)$. The traditional geometric interpretation of $X$ is to attach to each point $p$ of $U$ the vector $X(p)$ as an arrow based at $p$. This interpretation is appropriate, for example, if $X$ represents the velocity vector at each point in space of a time-independent fluid flow.

We have defined the term “path” in a metric space to mean a continuous function from a closed bounded interval of $\mathbb{R}^1$ into the metric space. The term curve in a metric space is used to refer to a continuous function from an open interval of $\mathbb{R}^1$ into the metric space.

A standard problem in connection with vector fields on an open subset $U$ of $\mathbb{R}^2$ is to try to draw curves within $U$ with the property that the tangent vector to the curve at any point matches the arrow for the vector field. An illustration occurs in Figure 4.2. This section abstracts and generalizes this kind of curve.

Let $X: U \rightarrow \mathbb{R}^n$ be a smooth vector field on $U$. A curve $c(t)$ is an integral curve for $X$ if $c$ is smooth and $c^{\prime}(t)=X(c(t))$ for all $t$ in the domain of c. Depending on one’s interpretation of the informal wording in the previous paragraph, the present definition is perhaps more demanding than the definition given for $\mathbb{R}^2$ above: the expression $c^{\prime}(t)$ involves both magnitude and direction, and the present definition insists that both ingredients match with $X(c(t))$, not just the direction.

## 数学代写|实分析作业代写Real analysis代考|Linear Equations and Systems, Wronskian

Recall from Section 1 that a linear ordinary differential equation is defined to be an equation of the type
$$a_n(t) y^{(n)}+a_{n-1}(t) y^{(n-1)}+\cdots+a_1(t) y^{\prime}+a_0(t) y=q(t)$$
with real or complex coefficients. The equation is homogeneous if $q$ is the 0 function, inhomogeneous in general. In order for the existence and uniqueness theorems of Section 1 to apply, we need to be able to solve for $y^{(n)}$ and have all coefficients be continuous afterward. Thus we assume that $a_n(t)=1$ and that $a_{n-1}(t), \ldots, a_0(t)$ and $q(t)$ are continuous on some open interval.

Even in simple cases, the theory is helped by converting a single equation to a system of first-order equations. In Section 1 we saw an indication that a way to make this conversion is to put

\begin{aligned} & y_1=y \ & y_1^{\prime}=y_2 \ & y_2=y^{\prime} \ & y_2^{\prime}=y_3 \ & \vdots \quad \text { and get } \ & y_{n-1}=y^{(n-2)} \ & y_{n-1}^{\prime}=y_n \ & y_n=y^{(n-1)} \ & y_n^{\prime}=-a_0(t) y_1-\cdots-a_{n-1} y_n+q(t) \text {. } \ & \end{aligned}
If we change the meaning of the symbol $y$ from a scalar-valued function to the vector-valued function $y=\left(y_1, \ldots, y_n\right)$, then we arrive at the system
$$y^{\prime}=A(t) y+Q(t),$$
where $A(t)$ is the $n$-by- $n$ matrix of continuous functions given by
$$A(t)=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & \vdots & \ 0 & 0 & 0 & \cdots & 1 \ -a_0(t) & -a_1(t) & -a_2(t) & \cdots & -a_{n-1}(t) \end{array}\right)$$
and $Q(t)$ is the $n$-component column vector of continuous functions given by
$$Q(t)=\left(\begin{array}{c} 0 \ 0 \ \vdots \ 0 \ q(t) \end{array}\right) .$$

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Linear Equations and Systems, Wronskian

$$a_n(t) y^{(n)}+a_{n-1}(t) y^{(n-1)}+\cdots+a_1(t) y^{\prime}+a_0(t) y=q(t)$$

\begin{aligned} & y_1=y \ & y_1^{\prime}=y_2 \ & y_2=y^{\prime} \ & y_2^{\prime}=y_3 \ & \vdots \quad \text { and get } \ & y_{n-1}=y^{(n-2)} \ & y_{n-1}^{\prime}=y_n \ & y_n=y^{(n-1)} \ & y_n^{\prime}=-a_0(t) y_1-\cdots-a_{n-1} y_n+q(t) \text {. } \ & \end{aligned}

$$y^{\prime}=A(t) y+Q(t),$$

$$A(t)=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \ 0 & 0 & 1 & \cdots & 0 \ \vdots & \vdots & \vdots & \vdots & \ 0 & 0 & 0 & \cdots & 1 \ -a_0(t) & -a_1(t) & -a_2(t) & \cdots & -a_{n-1}(t) \end{array}\right)$$

$$Q(t)=\left(\begin{array}{c} 0 \ 0 \ \vdots \ 0 \ q(t) \end{array}\right) .$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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