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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|空气动力学代写Aerodynamics代考|Entropy Variables

A particular choice of variables that symmetrizes the equations can be derived from functions of the entropy
$$S=\log \left(\frac{p}{\rho^\gamma}\right)=\log p-\gamma \log \rho .$$
The last equation of $(2.20)$ is equivalent to the statement that
$$\rho \frac{\partial S}{\partial t}+\rho u_i \frac{\partial S}{\partial x_i}=0,$$
which can be combined with the mass conservation equation multiplied by $S$,
$$S \frac{\partial \rho}{\partial t}+S \frac{\partial}{\partial x_i}\left(\rho u_i\right)=0,$$
to yield the entropy conservation law
$$\frac{\partial(\rho S)}{\partial t}+\frac{\partial\left(\rho u_i S\right)}{\partial x_i}=0 .$$
This is a special case of a generalized entropy function defined as follows. Given a system of conservation laws with the general form
$$\frac{\partial w}{\partial t}+\frac{\partial}{\partial x_i} \boldsymbol{f}_i(\boldsymbol{w})=0$$

suppose that we can find a scalar function $U(\boldsymbol{w})$ such that
$$\frac{\partial U}{\partial w} \frac{\partial f_i}{\partial w}=\frac{\partial G_i}{\partial w},$$
and $U(\boldsymbol{w})$ is a convex function of $\boldsymbol{w}$. Then $U(\boldsymbol{w})$ is an entropy function with an entropy flux $G_i(w)$ since multiplying (2.27) by $\frac{\partial U}{\partial w}$, we obtain
$$\frac{\partial U}{\partial w} \frac{\partial w}{\partial t}+\frac{\partial U}{\partial w} \frac{\partial f_i}{\partial w} \frac{\partial w}{\partial x_i}=0,$$
and using (2.28), this is equivalent to
$$\frac{\partial U(\boldsymbol{w})}{\partial t}+\frac{\partial G_i}{\partial w} \frac{\partial \boldsymbol{w}}{\partial x_i}=0,$$
which is in turn equivalent to the generalized entropy conservation law
$$\frac{\partial U(\boldsymbol{w})}{\partial t}+\frac{\partial}{\partial x_i} G_i(\boldsymbol{w})=0 .$$
Now, introduce dependent variables
$$\boldsymbol{v}^T=\frac{\partial U}{\partial w}$$

## 物理代写|空气动力学代写Aerodynamics代考|The Shock Jump Conditions for Gas Dynamics

The general jump conditions for a moving shock wave are
$$f_R-f_L=u_S\left(\boldsymbol{w}_R-\boldsymbol{w}_L\right),$$
where $w$ and $f(w)$ are the state and flux vectors, the subscripts $L$ and $R$ denote the conditions on the left and right side of the shock wave, and $u_S$ is the shock speed. Expanding (2.33) separately for mass, momentum, and energy:
\begin{aligned} \rho_L\left(u_L-u_S\right) &=\rho_R\left(u_R-u_S\right)=m \ \rho_L u_L\left(u_L-u_S\right)+p_L &=\rho_R u_R\left(u_R-u_S\right)+p_R \ \rho_L\left(i_L+\frac{u_L^2}{2}\right)\left(u_L-u_S\right)+u_L p_L &=\rho_R\left(i_R+\frac{u_R^2}{2}\right)\left(u_R-u_S\right)+u_R p_R, \end{aligned}
where $i$ is the specific internal energy and $m$ is the mass flow through the shock wave.
For a perfect gas,
$$i=\frac{p}{(\gamma-1) \rho} .$$
Equations (2.34) and (2.35) are called the mechanical conditions because they are independent of the thermodynamic properties and hold for any compressible fluid. They can be rearranged to give a variety of useful relations. Substituting (2.34) in (2.35),
$$m u_L+p_L=m u_R+p_R$$
or
$$m\left(u_L-u_S\right)+p_L=m\left(u_R-u_S\right)+p_R,$$ and hence conservation of momentum can be expressed as
$$\rho_L\left(u_L-u_S\right)^2+p_L=\rho_R\left(u_R-u_S\right)^2+p_R=P,$$
where $P$ is the momentum flux.

# 空气动力学代考

## 物理代写|空气动力学代写空气动力学代考|熵变量

$$S=\log \left(\frac{p}{\rho^\gamma}\right)=\log p-\gamma \log \rho .$$

$$\rho \frac{\partial S}{\partial t}+\rho u_i \frac{\partial S}{\partial x_i}=0,$$
，它可以与质量守恒方程乘以$S$结合，
$$S \frac{\partial \rho}{\partial t}+S \frac{\partial}{\partial x_i}\left(\rho u_i\right)=0,$$

$$\frac{\partial(\rho S)}{\partial t}+\frac{\partial\left(\rho u_i S\right)}{\partial x_i}=0 .$$

$$\frac{\partial w}{\partial t}+\frac{\partial}{\partial x_i} \boldsymbol{f}_i(\boldsymbol{w})=0$$

$$\frac{\partial U}{\partial w} \frac{\partial f_i}{\partial w}=\frac{\partial G_i}{\partial w},$$

$$\frac{\partial U}{\partial w} \frac{\partial w}{\partial t}+\frac{\partial U}{\partial w} \frac{\partial f_i}{\partial w} \frac{\partial w}{\partial x_i}=0,$$
，并使用(2.28)，这等价于
$$\frac{\partial U(\boldsymbol{w})}{\partial t}+\frac{\partial G_i}{\partial w} \frac{\partial \boldsymbol{w}}{\partial x_i}=0,$$
，这又等价于广义熵守恒定律
$$\frac{\partial U(\boldsymbol{w})}{\partial t}+\frac{\partial}{\partial x_i} G_i(\boldsymbol{w})=0 .$$

$$\boldsymbol{v}^T=\frac{\partial U}{\partial w}$$

## 物理代写|空气动力学代写空气动力学代考|气体动力学的激波跳跃条件

$$f_R-f_L=u_S\left(\boldsymbol{w}_R-\boldsymbol{w}_L\right),$$
where $w$ 和 $f(w)$ 状态向量和通量向量是下标吗 $L$ 和 $R$ 表示冲击波左右两侧的条件，和 $u_S$ 是冲击速度。分别为质量、动量和能量展开(2.33):
\begin{aligned} \rho_L\left(u_L-u_S\right) &=\rho_R\left(u_R-u_S\right)=m \ \rho_L u_L\left(u_L-u_S\right)+p_L &=\rho_R u_R\left(u_R-u_S\right)+p_R \ \rho_L\left(i_L+\frac{u_L^2}{2}\right)\left(u_L-u_S\right)+u_L p_L &=\rho_R\left(i_R+\frac{u_R^2}{2}\right)\left(u_R-u_S\right)+u_R p_R, \end{aligned}
where $i$ 比热力学能和 $m$ 是穿过激波的质量流。对于完美气体，
$$i=\frac{p}{(\gamma-1) \rho} .$$(2.34)和(2.35)式称为力学条件，因为它们与热力学性质无关，对任何可压缩流体都成立。它们可以重新排列，以提供各种有用的关系。将(2.34)替换为(2.35)，
$$m u_L+p_L=m u_R+p_R$$

$$m\left(u_L-u_S\right)+p_L=m\left(u_R-u_S\right)+p_R,$$ 因此动量守恒可以表示为
$$\rho_L\left(u_L-u_S\right)^2+p_L=\rho_R\left(u_R-u_S\right)^2+p_R=P,$$
where $P$ 动量通量。

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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。

assignmentutor™您的专属作业导师
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