数学代写|有限元方法代写Finite Element Method代考|GENG5514

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数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

In case the material is elastically isotropic and the stress and strain tensors are symmetric the material behavior can be characterized with two material constants,
E: Elastic modulus or Young’s modulus
v: Poisson’s ratio
For a three-dimensional problem, it can be shown that the following relationships exist between the stresses and strains,
\begin{aligned} \varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \tau_{x y} & =G \gamma_{x y} \ \tau_{y z} & =G \gamma_{y z} \ \tau_{z x} & =G \gamma_{z x} \end{aligned}
where shear modulus $G=E / 2(1+v)$.

Note that Eq. (2.61a) can be inverted and expressed as follows:
\begin{aligned} \sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \ \sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \ \sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \ \tau_{x y} & =\mu \gamma_{x y} \ \tau_{y z} & =\mu \gamma_{y z} \ \tau_{z x} & =\mu \gamma_{z x} \end{aligned}
where, the Lamé constants are defined as follows:
\begin{aligned} & \lambda=\frac{v E}{(1+v)(1-2 v)} \ & \mu=G \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

Thermal stress in a one-dimensional problem: Consider a long and slender bar of length $L$ and initial temperature $T^{(0)}$. If the temperature of the bar is changed by $\Delta T$, material points in the bar would experience thermal strain proportional to the temperature change,
$$\varepsilon^{(t h)}=\alpha \Delta T$$
the proportionality constant $\alpha$ is a material property known as the coefficient of thermal expansion with units of $\mathrm{K}^{-1}$ or $\left({ }^{\circ} \mathrm{C}\right)^{-1}$. If the bar is not constrained on its ends, its length will change by an amount,
$$\Delta L=\int_0^L \alpha \Delta T d x$$
but no internal stress will develop.

On the other hand if both ends of the bar are constrained, internal forces and hence stress will develop in the bar. If such constraint conditions exist, the thermal stress in the bar can be found from Hooke’s law as follows:
$$\sigma^{(t h)}=E \alpha \Delta T$$
Next, consider a constrained bar subjected to external forces and change of temperature. The total strain in this bar can be found by using the superposition of the mechanical component of the strain and the thermal strain,
$$\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T$$
The inverse of this relation gives the corresponding total stress,
$$\sigma=E(\varepsilon-\alpha \Delta T)$$
Generalized stress-strain relations with thermal effects: For materials with isotropic material properties temperature change only causes normal strain in the material. The stress-strain relations for a three-dimensional isotropic material subjected to a temperature change $\Delta T$ are expressed as follows [8]:
\begin{aligned} \varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \gamma_{x y} & =\frac{\tau_{x y}}{G} \ \gamma_{y z} & =\frac{\tau_{y z}}{G} \ \gamma_{z x} & =\frac{\tau_{z x}}{G} \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law for isotropic materials with symmetric stress and strain tensors

E:弹性模量或杨氏模量
v:泊松比

\begin{aligned} \varepsilon_{x x} & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y} & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z} & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \tau_{x y} & =G \gamma_{x y} \ \tau_{y z} & =G \gamma_{y z} \ \tau_{z x} & =G \gamma_{z x} \end{aligned}

\begin{aligned} \sigma_{x x} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{x x} \ \sigma_{y y} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{y y} \ \sigma_{z z} & =\lambda\left(\varepsilon_{x x}+\varepsilon_{y y}+\varepsilon_{z z}\right)+2 \mu \varepsilon_{z z} \ \tau_{x y} & =\mu \gamma_{x y} \ \tau_{y z} & =\mu \gamma_{y z} \ \tau_{z x} & =\mu \gamma_{z x} \end{aligned}

\begin{aligned} & \lambda=\frac{v E}{(1+v)(1-2 v)} \ & \mu=G \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Effects of initial stress/strain and thermal strain

$$\varepsilon^{(t h)}=\alpha \Delta T$$

$$\Delta L=\int_0^L \alpha \Delta T d x$$

$$\sigma^{(t h)}=E \alpha \Delta T$$

$$\varepsilon=\frac{\sigma}{E}+\varepsilon^{(t h)}=\frac{\sigma}{E}+\alpha \Delta T$$

$$\sigma=E(\varepsilon-\alpha \Delta T)$$

\begin{aligned} \varepsilon_{x x}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{x x}-v\left(\sigma_{y y}+\sigma_{z z}\right)\right] \ \varepsilon_{y y}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{y y}-v\left(\sigma_{z z}+\sigma_{x x}\right)\right] \ \varepsilon_{z z}-\alpha \Delta T & =\frac{1}{E}\left[\sigma_{z z}-v\left(\sigma_{x x}+\sigma_{y y}\right)\right] \ \gamma_{x y} & =\frac{\tau_{x y}}{G} \ \gamma_{y z} & =\frac{\tau_{y z}}{G} \ \gamma_{z x} & =\frac{\tau_{z x}}{G} \end{aligned}

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|有限元方法代写Finite Element Method代考|ENGR7961

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

An elastic deformation should not cause holes in a deformable body that does not have any holes before deformation. Moreover, no material overlap should be predicted by the displacement field. The strain compatibility conditions ensure that these constraints are satisfied [7].

In a planar deformation, where $u_x=u_x(x, y), u_y=u_y(x, y)$ and $u_z=0$, consider the following combination of the strains,
$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

Using the definitions given in Eq. (2.47) we will find,
$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$
Thus we see that the relationship (a) must be equal to zero. This is the strain compatibility equation for a two-dimensional deformation in the $x, y$ plane, which imposes a specific relationship between the strains and the strain-displacement relationships.

For three-dimensional deformations where $u_x=u_x(x, y, z), u_y=u_y(x, y, z)$ and $u_z=u_z(x, y, z)$ there are a total of six strain compatibility conditions. These can be found as follows:
\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

In previous sections it was indicated that, in general, the stress and strain tensors at a point have nine independent components each, if we do not take into account the symmetries. Therefore, the possibility exists for all of these 18 components to be interrelated. In it most general form, the linear elastic constitutive law, also known as generalized Hooke’s law, can be expressed as follows:
$$\sigma_{i j}=c_{i j r s} \varepsilon_{r s}$$
where the subscripts $i, j, r, s=x, y, z$ and the coefficients $c_{i j r s}$ are empirically determined. Note that the tensor notation is used in expressing Eq. (2.57) where $\sigma$ and $\varepsilon$ are second order tensors and $c_{i j r s}$ is a fourth order tensor [7]. Repeated indices imply summation, such that for $\sigma_{x x}$ the most general form of the Hooke’s law would be,
\begin{aligned} \sigma_{x x}= & c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x x z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \ & +c_{x x z y} \gamma_{z y}+c_{x x z z} \varepsilon_{z z} \end{aligned}

It can easily be deduced that 81 material properties would be required in case of an anisotropic material with no-symmetries in the strain and stress tensors. In matrix notation, Eq. (2.57) can be expressed as follows:
$${\sigma}=[E]{\varepsilon}$$
where $[E]$ is an $81 \times 81$ elasticity matrix and ${\sigma}$ and ${\varepsilon}$ are $9 \times 1$ vectors.

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$

\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

$$\sigma_{i j}=c_{i j r s} \varepsilon_{r s}$$

\begin{aligned} \sigma_{x x}= & c_{x x x x} \varepsilon_{x x}+c_{x x x y} \gamma_{x y}+c_{x x x z} \gamma_{x z}+c_{x x y x} \gamma_{y x}+c_{x x y y} \varepsilon_{y y}+c_{x x y z} \gamma_{y z}+c_{x x z x} \gamma_{z x} \ & +c_{x x z y} \gamma_{z y}+c_{x x z z} \varepsilon_{z z} \end{aligned}

$${\sigma}=[E]{\varepsilon}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|有限元方法代写Finite Element Method代考|MECH3300

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

An elastic deformation should not cause holes in a deformable body that does not have any holes before deformation. Moreover, no material overlap should be predicted by the displacement field. The strain compatibility conditions ensure that these constraints are satisfied [7].

In a planar deformation, where $u_x=u_x(x, y), u_y=u_y(x, y)$ and $u_z=0$, consider the following combination of the strains,
$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

Using the definitions given in Eq. (2.47) we will find,
$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$
Thus we see that the relationship (a) must be equal to zero. This is the strain compatibility equation for a two-dimensional deformation in the $x, y$ plane, which imposes a specific relationship between the strains and the strain-displacement relationships.

For three-dimensional deformations where $u_x=u_x(x, y, z), u_y=u_y(x, y, z)$ and $u_z=u_z(x, y, z)$ there are a total of six strain compatibility conditions. These can be found as follows:
\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

As stated in the introduction to Section 2.2, when a deformable body is subjected to external effects such as external forces and/or imposed displacements on its boundary, its shape will change and internal forces will develop throughout its volume. The level of deformation for given external effects depends on the material of the deformable body. Constitutive relations are empirically obtained, material specific relationships between the stress and the strain in the body. Here we are primarily interested in linear elastic relationships.

The deformation behavior of a specific material is determined experimentally. These experiments are designed such that only one of the stress components and the corresponding strain dominates the problem. This state is known as a simpleloading state.

For linear, isotropic materials tensile loading of a slender test specimen, i.e., the simple-tension test, reveals two fundamental material properties. The relationship between the normal stress and the normal strain is found by conducting a simple-tension test, as follows:
$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$
where $E$ is the elastic modulus of the material, also referred to as the Young’s modulus. The relationship between the longitudinal strain $\varepsilon_l$ and the transverse strain $\varepsilon_t$ represents the Poisson’s ratio, the second material property,
$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$
The simple-shear test reveals the relationship between the shear strain and the shear stress,
$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$
where $G$ is the shear modulus, or modulus of rigidity. For a linear, elastic, isotropic material the following relationship holds:
$$G=\frac{E}{2(1+v)}$$

数学代写|有限元方法代写Finite Element Method代考|Strain compatibility conditions

$$\frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}$$

$$\frac{\partial^3 u_y}{\partial x^2 \partial y}+\frac{\partial^3 u_x}{\partial y^2 \partial x}-\frac{\partial^2}{\partial x \partial y}\left(\frac{\partial u_y}{\partial x}+\frac{\partial u_x}{\partial y}\right)=0$$

\begin{aligned} & \frac{\partial^2 \varepsilon_{y y}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial y^2}-\frac{\partial^2 \gamma_{x y}}{\partial x \partial y}=0 \ & \frac{\partial^2 \varepsilon_{y y}}{\partial z^2}+\frac{\partial^2 \varepsilon_{z z}}{\partial y^2}-\frac{\partial^2 \gamma_{y z}}{\partial z \partial y}=0 \ & \frac{\partial^2 \varepsilon_{z z}}{\partial x^2}+\frac{\partial^2 \varepsilon_{x x}}{\partial z^2}-\frac{\partial^2 \gamma_{z y}}{\partial x \partial z}=0 \ & 2 \frac{\partial^2 \varepsilon_{x x}}{\partial y \partial z}=\frac{\partial}{\partial x}\left(-\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}\right) \ & 2 \frac{\partial^2 \varepsilon_{y y}}{\partial z \partial x}=\frac{\partial}{\partial y}\left(-\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}\right) \ & 2 \frac{\partial^2 \varepsilon_{z z}}{\partial x \partial y}=\frac{\partial}{\partial z}\left(-\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}\right) \end{aligned}

数学代写|有限元方法代写Finite Element Method代考|Generalized Hooke’s law

$$\sigma_{i i}=E \varepsilon_{i i} \quad \text { for } \quad i=x, y, z$$

$$v=-\frac{\varepsilon_t}{\varepsilon_l}$$

$$\tau_{i j}=G \gamma_{i j} \quad \text { for } \quad i, j=x, y, z \quad \text { and } \quad i \neq j$$

$$G=\frac{E}{2(1+v)}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|信息论代写information theory代考|FEO3350

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

In this section we consider a special case of a system of interacting particles. We start with an ideal gas-i.e. system for which we can neglect all intermolecular interactions. Strictly speaking, such a system does not exist. However, if the gas is very dilute such that the average intermolecular distance is very large the system behaves as if there are no interactions among the particle.

Next, we increase the density of the particles. At first we shall find that pairinteractions affect the thermodynamics of the system. Increasing further the density, triplets, quadruplets, and so on interactions, will also affect the behavior of the system. In the following we provide a very brief description of the first order deviation from ideal gas; systems for which one must take into account pair-interactions but neglect triplet and higher order interactions. The reader who is not interested in the details of the derivation can go directly to the result in Eq. (2.51) and the following analysis of the MI.

We start with the general configurational PF of the system, Eq. (2.31) which we rewrite in the form:
$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$
where $U_{i j}$ is the pair potential between particles $i$ and $j$. It is assumed that the total potential energy is pairwise additive.
Define the so-called Mayer $f$-function, by:
$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$
We can rewrite $Z_N$ as:
$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$
Neglecting all terms beyond the first sum, we obtain:
$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

In this section, we shall discuss the entropy-changes associated with phase transitions. Here, by entropy we mean thermodynamic entropy, the units of which are cal/(deg $\mathrm{mol}$ ). However, as we have seen in Chap. 5 of Ben-Naim [1]. The entropy is up to a multiplicative constant an SMI defined on the distribution of locations and velocities (or momenta) of all particles in the system at equilibrium. To convert from entropy to SMI one has to divide the entropy by the factor $k_B \log _e 2$, where $k_B$ is the Boltzmann constant, and $\log _e 2$ is the natural $\log$ arithm of 2 , which we denote by $\ln 2$. Once we do this conversion from entropy to SMI we obtain the SMI in units of bits. In this section we shall discuss mainly the transitions between gases, liquids and solids. Figure 2.9 shows a typical phase diagram of a one-component system. For more details on phase diagrams, see Ben-Naim and Casadei [8].

It is well-known that solid has a lower entropy than liquid, and liquid has a lower entropy of a gas. These facts are usually interpreted in terms of order-disorder. This interpretation of entropy is invalid; more on this in Ben-Naim [6]. Although, it is true that a solid is viewed as more ordered than liquid, it is difficult to argue that a liquid is more ordered or less ordered than a gas.

In the following we shall interpret entropy as an SMI, and different entropies in terms of different MI due to different intermolecular interactions. We shall discuss changes of phases at constant temperature. Therefore, all changes in SMI (hence, in entropy) will be due to locational distributions; no changes in the momenta distribution.

The line SG in Fig. 2.9 is the line along in which solid and gas coexist. The slope of this curve is given by:
$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$
In the process of sublimation ( $s$, the entropy-change and the volume change for both are always positive. We denoted by $\Delta V_s$ the change in the volume of one mole of the substance, when it is transferred from the solid to the gaseous phase. This volume change is always positive. The reason is that a mole of the substance occupies a much larger volume in the gaseous phase than in the liquid phase (at the same temperature and pressure).

The entropy-change $\Delta S_s$ is also positive. This entropy-change is traditionally interpreted in terms of transition from an ordered phase (solid) to a disordered (gaseous) phase. However, the more correct interpretation is that the entropy-change is due to two factors; the huge increase in the accessible volume available to each particle and the decrease in the extent of the intermolecular interaction. Note that the slope of the SG curve is quite small (but positive) due to the large $\Delta V_s$.

信息论代写

数学代写|信息论代写information theory代考|The SMI of a System of Interacting Particles in Pairs Only

$$Z_N=\int d R^N \prod_{i<j} \exp \left[-\beta U_{i j}\right]$$

$$f_{i j}=\exp \left(-\beta U_{i j}\right)-1$$

$$Z_N=\int d R^N \prod_{i<j}\left(f_{i j}+1\right)=\int d R^N\left[1+\sum_{i<j} f_{i j}+\sum f_{i j} f_{j k}+\cdots\right]$$

$$Z_N=V^N+\frac{N(N-1)}{2} \int f_{12} d R^N=V^N+\frac{N(N-1)}{2} V^{N-2} \int f_{12} d R_1 d R_2$$

数学代写|信息论代写information theory代考|Entropy-Change in Phase Transition

$$\left(\frac{d P}{d T}\right)_{e q}=\frac{\Delta S_s}{\Delta V_s}$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|信息论代写information theory代考|EE430

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

数学代写|信息论代写information theory代考|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

The final step is to proceed from a single particle in a box, to $N$ independent particles in a box of volume $V$, Fig. 2.4.

We say that we know the microstate of the particle, when we know the location $(x, y, z)$, and the momentum $\left(p_x, p_y, p_z\right)$ of one particle within the box. For a system of $N$ independent particles in a box, we can write the SMI of the system as $N$ times the SMI of one particle, i.e., we write:
$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$
This is the SMI for $N$ independent particles. In reality, there could be correlation among the microstates of all the particles. We shall mention here correlations due to the indistinguishability of the particles, and correlations is due to intermolecular interactions among all the particles. We shall discuss these two sources of correlation separately. Recall that the microstate of a single particle includes the location and the momentum of that particle. Let us focus on the location of one particle in a box of volume $V$. We write the locational SMI as:
$$H_{\max }(\text { location })=\log V$$
For $N$ independent particles, we write the locational SMI as:
$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$
Since in reality, the particles are indistinguishable, we must correct Eq. (2.22). We define the mutual information corresponding to the correlation between the particles as:

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$
Hence, instead of (2.22), for the SMI of $N$ indistinguishable particles, will write:
$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$
A detailed justification for introducing $\ln N$ ! as a correction due to indistinguishability of the particle is discussed in Sect. 5.2 of Ben-Naim [1]. Here we write the final result for the SMI of $N$ indistinguishable (but non-interacting) particles as:
$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

In this section we derive the most general relationship between the SMI (or the entropy) of a system of interacting particles, and the corresponding mutual information (MI). Later on in this chapter we shall apply this general result to some specific cases. The implication of this result is very important in interpreting the concept of entropy in terms of SMI. In other words, the “informational interpretation” of entropy is effectively extended for all systems of interacting particles at equilibrium.
We start with some basic concepts from classical statistical mechanics [7]. The classical canonical partition function (PF) of a system characterized by the variable $T, V, N$, is:
$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$
where $\Lambda^3$ is called the momentum partition function (or the de Broglie wavelength), and $Z_N$ is the configurational PF of the system”
$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$
Here, $U_N\left(R^N\right)$ is the total interaction energy among the $N$ particles at a configuration $R^N=R_1, \cdots, R_N$. Statistical thermodynamics provides the probability density for finding the particles at a specific configuration $R^N=R_1, \cdots, R_N$, which is:
$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$
where $\beta=\left(k_B T\right)^{-1}$ and $T$ the absolute temperature. In the following we chose $k_B=1$. This will facilitate the connection between the entropy-change and the change in the SMI. When there are no intermolecular interactions (ideal gas), the configurational $\mathrm{PF}$ is $Z_N=V^N$, and the corresponding partition function is reduced to:
$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$
Next we define the change in the Helmholtz energy $(A)$ due to the interactions as:
$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$
This change in Helmholtz energy corresponds to the process of “turning-on” the interaction among all the particles at constant $(T, V, N)$, Fig. 2.5.
The corresponding change in the entropy is:
\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}
We now substitute $U_N\left(R^N\right)$ from (2.36) into (2.35) to obtain the expression for the change in entropy corresponding to “turning on” the interactions:
$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

信息论代写

数学代写|信息论代写information theory代考|The Forth Step: The SMI of Locations and Momentaof N Independent Particles in a Box of Volume V.Adding a Correction Due to Indistinguishabilityof the Particles

$$\mathrm{SMI}(N \text { independent particles })=N \times \mathrm{SMI} \text { (one particle) }$$

$$H_{\max }(\text { location })=\log V$$

$$H_{\max } \text { (locations of N particles) }=\sum_{i=1}^N H_{\max }(\text { one particle })$$

$$I(1 ; 2 ; \ldots ; N)=\ln N !$$

$$H(\text { Nparticles })=\sum_{i=1}^N H(\text { oneparticle })-\ln N !$$

$$H(N \text { indistinguishable particles })=N \log V\left(\frac{2 \pi m e k_B T}{h^2}\right)^{3 / 2}-\log N !$$

数学代写|信息论代写information theory代考|The Entropy of a System of Interacting Particles. Correlations Due to Intermolecular Interactions

$$Q(T, V, N)=\frac{Z_N}{N ! \Lambda^{3 N}}$$

$$Z_N=\int \cdots \int d R^N \exp \left[-\beta U_N\left(R^N\right)\right]$$

$$P\left(R^N\right)=\frac{\exp \left[-\beta U_N\left(R^N\right)\right]}{Z_N}$$

$$Q^{i g}(T, V, N)=\frac{V^N}{N ! \Lambda^{3 N}}$$

$$\Delta A=A-A^{i g}=-T \ln \frac{Q(T, V, N)}{Q^{i g}(T, V, N)}=-T \ln \frac{Z_N}{V^N}$$

\begin{aligned} \Delta S & =-\frac{\partial \Delta A}{\partial T}=\ln \frac{Z_N}{V^N}+T \frac{1}{Z_N} \frac{\partial Z_N}{\partial T} \ & =\ln Z_N-N \ln \mathrm{V}+\frac{1}{T} \int d R^N P\left(R^N\right) U_N\left(R^N\right) \end{aligned}

$$\Delta S=-N \ln V-\int P\left(R^N\right) \ln P\left(R^N\right) d R^N$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|信息论代写information theory代考|COMP2610

statistics-lab™ 为您的留学生涯保驾护航 在代写信息论information theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写信息论information theory代写方面经验极为丰富，各种代写信息论information theory相关的作业也就用不着说。

数学代写|信息论代写information theory代考|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

If the location and the momentum (or velocity) of the particles were independent events, then the joint SMI of location and momentum would be the sum of the two SMIs in Eqs. (2.4) and (2.12). Therefore, for this case we write:
\begin{aligned} H_{\max }(\text { location and momentum }) & =H_{\max }(\text { location })+H_{\max }(\text { momentum }) \ & =\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h_x h_p}\right] \end{aligned}
It should be noted that in the very writing of Eq. (2.14), the assumption is made that the location and the momentum of the particle are independent. However, quantum mechanics imposes restriction on the accuracy in determining both the location $x$ and the corresponding momentum $p_x$. Originally, the two quantities $h_x$ and $h_p$ that we defined above, were introduced because we did not care to determine the location and the momentum with an accuracy better than $h_x$ and $h_p$, respectively. Now, we must acknowledge that quantum mechanics imposes upon us the uncertainty condition, about the accuracy with which we can determine simultaneously both the location and the corresponding momentum of a particle. This means that in Eq. (2.14), $h_x$ and $h_p$ cannot both be arbitrarily small; their product must be of the order of Planck constant $h=6.626 \times 10^{-34} \mathrm{Js}$. Therefore, we introduce a new parameter $h$, which replaces the product:
$$h_x h_p \approx h$$
Accordingly, we modify Eq. (2.14) to:
$$H_{\max }(\text { location and momentum })=\log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

Figure 2.3 shows one simple particle in a cubic box of volume $V$.
To proceed from the 1D to the 3D system, we assume that the locations of the particle along the three axes $x, y$ and $z$ are independent. With this assumption, we can write the SMI of the location of the particle in a cube of edges $L$, as a sum of the SMI along $x, y$, and $z$, i.e.
$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$
We can do the same for the momentum of the particle if we assume that the momentum (or the velocity) along the three axes $x, y$ and $z$ are independent. Hence, we can write the SMI of the momentum as:
$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$
We can now combine the SMI of the locations and momenta of one particle in a box of volume $V$, taking into account the uncertainty principle, to obtain the result:
$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

信息论代写

数学代写|信息论代写information theory代考|Third Step: Combining the SMI for the Location and Momentum of a Particle in a $1 D$ System. Addition of Correction Due to Uncertainty

$$H[f(x)]=-\int f(x) \log f(x) d x$$

$$f_{e q}(x)=\frac{1}{L}$$

$$H(\text { locations in } 1 D)=\log L$$

$$H\left(\text { locations in 1D) }=\log L-\log h_x\right.$$

数学代写|信息论代写information theory代考|The SMI of One Particle in a Box of Volume $\mathrm{V}$

$$H(\text { location in } 3 \mathrm{D})=3 H_{\max } \text { (location in 1D) }$$

$$H_{\max }(\text { momentum in } 3 \mathrm{D})=3 H_{\max }(\text { momentum in 1D) }$$

$$H_{\max }(\text { location and momentum in } 3 \mathrm{D})=3 \log \left[\frac{L \sqrt{2 \pi e m k_B T}}{h}\right]$$

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

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。

数学代写|图论作业代写Graph Theory代考|MTH607

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数学代写|图论作业代写Graph Theory代考|Ramsey properties and connectivity

According to Ramsey’s theorem, every large enough graph $G$ has a very dense or a very sparse induced subgraph of given order, a $K^r$ or $\overline{K^r}$. If we assume that $G$ is connected, we can say a little more:

Proposition 9.4.1. For every $r \in \mathbb{N}$ there is an $n \in \mathbb{N}$ such that every connected graph of order at least $n$ contains $K^r, K_{1, r}$ or $P^r$ as an induced subgraph.

Proof. Let $d+1$ be the Ramsey number of $r$, let $n:=\frac{d}{d-2}(d-1)^r$, and let $G$ be a graph of order at least $n$. If $G$ has a vertex $v$ of degree at least $d+1$ then, by Theorem 9.1.1 and the choice of $d$, either $N(v)$ induces a $K^r$ in $G$ or ${v} \cup N(v)$ induces a $K_{1, r}$. On the other hand, if $\Delta(G) \leqslant d$, then by Proposition 1.3.3 $G$ has radius $>r$, and hence contains two vertices at a distance $\geqslant r$. Any shortest path in $G$ between these two vertices contains a $P^r$.

In principle, we could now look for a similar set of ‘unavoidable’ $k$-connected subgraphs for any given connectivity $k$. To keep thse ‘unavoidable sets’ small, it helps to relax the containment relation from ‘induced subgraph’ for $k=1$ (as above) to ‘topological minor’ for $k=2$, and on to ‘minor’ for $k=3$ and $k=4$. For larger $k$, no similar results are known.

Proposition 9.4.2. For every $r \in \mathbb{N}$ there is an $n \in \mathbb{N}$ such that every 2-connected graph of order at least $n$ contains $C^r$ or $K_{2, r}$ as a topological minor.

Proof. Let $d$ be the $n$ associated with $r$ in Proposition 9.4.1, and let $G$ be a 2-connected graph with at least $\frac{d}{d-2}(d-1)^r$ vertices. By Proposition 1.3.3, either $G$ has a vertex of degree $>d$ or $\operatorname{diam} G \geqslant \operatorname{rad} G>r$.

In the latter case let $a, b \in G$ be two vertices at distance $>r$. By Menger’s theorem (3.3.6), $G$ contains two independent $a-b$ paths. These form a cycle of length $>r$.

数学代写|图论作业代写Graph Theory代考|Simple sufficient conditions

What kind of condition might be sufficient for the existence of a Hamilton cycle in a graph $G$ ? Purely global assumptions, like high edge density, will not be enough: we cannot do without the local property that every vertex has at least two neighbours. But neither is any large (but constant) minimum degree sufficient: it is easy to find graphs without a Hamilton cycle whose minimum degree exceeds any given constant bound.
The following classic result derives its significance from this background:

Theorem 10.1.1. (Dirac 1952)
Every graph with $n \geqslant 3$ vertices and minimum degree at least $n / 2$ has a Hamilton cycle.

Proof. Let $G=(V, E)$ be a graph with $|G|=n \geqslant 3$ and $\delta(G) \geqslant n / 2$. Then $G$ is connected: otherwise, the degree of any vertex in the smallest component $C$ of $G$ would be less than $|C| \leqslant n / 2$.

Let $P=x_0 \ldots x_k$ be a longest path in $G$. By the maximality of $P$, all the neighbours of $x_0$ and all the neighbours of $x_k$ lie on $P$. Hence at least $n / 2$ of the vertices $x_0, \ldots, x_{k-1}$ are adjacent to $x_k$, and at least $n / 2$ of these same $k<n$ vertices $x_i$ are such that $x_0 x_{i+1} \in E$. By the pigeon hole principle, there is a vertex $x_i$ that has both properties, so we have $x_0 x_{i+1} \in E$ and $x_i x_k \in E$ for some $i<k$ (Fig. 10.1.1).

We claim that the cycle $C:=x_0 x_{i+1} P x_k x_i P x_0$ is a Hamilton cycle of $G$. Indeed, since $G$ is connected, $C$ would otherwise have a neighbour in $G-C$, which could be combined with a spanning path of $C$ into a path longer than $P$.

图论代考

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。

数学代写|图论作业代写Graph Theory代考|Math780

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

数学代写|图论作业代写Graph Theory代考|Ramsey numbers

Ramsey’s theorem may be rephrased as follows: if $H=K^r$ and $G$ is a graph with sufficiently many vertices, then either $G$ itself or its complement $\bar{G}$ contains a copy of $H$ as a subgraph. Clearly, the same is true for any graph $H$, simply because $H \subseteq K^h$ for $h:=|H|$.

However, if we ask for the least $n$ such that every graph $G$ of order $n$ has the above property – this is the Ramsey number $R(H)$ of $H$-then the above question makes sense: if $H$ has only few edges, it should embed more easily in $G$ or $\bar{G}$, and we would expect $R(H)$ to be smaller than the Ramsey number $R(h)=R\left(K^h\right)$.

A little more generally, let $R\left(H_1, H_2\right)$ denote the least $n \in \mathbb{N}$ such that $H_1 \subseteq G$ or $H_2 \subseteq \bar{G}$ for every graph $G$ of order $n$. For most graphs $H_1, H_2$, only very rough estimates are known for $R\left(H_1, H_2\right)$. Interestingly, lower bounds given by random graphs (as in Theorem 11.1.3) are often sharper than even the best bounds provided by explicit constructions.

The following proposition describes one of the few cases where exact Ramsey numbers are known for a relatively large class of graphs:

Proposition 9.2.1. Let $s, t$ be positive integers, and let $T$ be a tree of order $t$. Then $R\left(T, K^s\right)=(s-1)(t-1)+1$.

Proof. The disjoint union of $s-1$ graphs $K^{t-1}$ contains no copy of $T$, while the complement of this graph, the complete $(s-1)$-partite graph $K_{t-1}^{s-1}$, does not contain $K^s$. This proves $R\left(T, K^s\right) \geqslant(s-1)(t-1)+1$.
Conversely, let $G$ be any graph of order $n=(s-1)(t-1)+1$ whose complement contains no $K^s$. Then $s>1$, and in any vertex colouring of $G$ (in the sense of Chapter 5) at most $s-1$ vertices can have the same colour. Hence, $\chi(G) \geqslant\lceil n /(s-1)\rceil=t$. By Corollary $5.2 .3, G$ has a subgraph $H$ with $\delta(H) \geqslant t-1$, which by Corollary 1.5 .4 contains a copy of $T$.

数学代写|图论作业代写Graph Theory代考|Induced Ramsey theorems

Ramsey’s theorem can be rephrased as follows. For every graph $H=K^r$ there exists a graph $G$ such that every 2-colouring of the edges of $G$ yields a monochromatic $H \subseteq G$; as it turns out, this is witnessed by any large enough complete graph as $G$. Let us now change the problem slightly and ask for a graph $G$ in which every 2-edge-colouring yields a monochromatic induced $H \subseteq G$, where $H$ is now an arbitrary given graph.

This slight modification changes the character of the problem dramatically. What is needed now is no longer a simple proof that $G$ is ‘big enough’ (as for Theorem 9.1.1), but a careful construction: the construction of a graph that, however we bipartition its edges, contains an induced copy of $H$ with all edges in one partition class. We shall call such a graph a Ramsey graph for $H$.

The fact that such a Ramsey graph exists for every choice of $H$ is one of the fundamental results of graph Ramsey theory. It was proved around 1973 , independently by Deuber, by Erdős, Hajnal \& Pósa, and by Rödl.

Theorem 9.3.1. Every graph has a Ramsey graph. In other words, for every graph $H$ there exists a graph $G$ that, for every partition $\left{E_1, E_2\right}$ of $E(G)$, has an induced subgraph $H$ with $E(H) \subseteq E_1$ or $E(H) \subseteq E_2$.

We give two proofs. Each of these is highly individual, yet each offers a glimpse of true Ramsey theory: the graphs involved are used as hardly more than bricks in the construction, but the edifice is impressive.

图论代考

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。

数学代写|图论作业代写Graph Theory代考|MATH7331

statistics-lab™ 为您的留学生涯保驾护航 在代写图论Graph Theory方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写图论Graph Theory代写方面经验极为丰富，各种代写图论Graph Theory相关的作业也就用不着说。

数学代写|图论作业代写Graph Theory代考|The topological end space

In this last section we shall develop a deeper understanding of the global structure of infinite graphs, especially locally finite ones, that can be attained only by studying their ends. This structure is intrinsically topological, but no more than the most basic concepts of point-set topology will be needed.

Our starting point will be to make precise the intuitive idea that the ends of a graph are the ‘points at infinity’ to which its rays converge. To do so, we shall define a topological space $|G|$ associated with a graph $G=(V, E, \Omega)$ and its ends. ${ }^8$ By considering topological versions of paths, cycles and spanning trees in this space, we shall then be able to extend to infinite graphs some parts of finite graph theory that would not otherwise have infinite counterparts (see the notes for more examples). Thus, the ends of an infinite graph turn out to be more than a curious new phenomenon: they form an integral part of the picture, without which it cannot be properly understood.

To build the space $|G|$ formally, we start with the set $V \cup \Omega$. For every edge $e=u v$ we add a set $\dot{e}=(u, v)$ of continuum many points, making these sets $\ddot{e}$ disjoint from each other and from $V \cup \Omega$. We then choose for each $e$ some fixed bijection between $\dot{e}$ and the real interval $(0,1)$, and extend this bijection to one between $[u, v]:={u} \cup \grave{e} \cup{v}$ and $[0,1]$. This bijection defines a metric on $[u, v]$; we call $[u, v]$ a topological edge with inner points $x \in \dot{e}$. Given any $F \subseteq E$ we write $\stackrel{\circ}{F}:=\bigcup{\dot{e} \mid e \in F}$.

When we speak of a ‘graph’ $H \subseteq G$, we shall often also mean its corresponding point set $V(H) \cup \tilde{E}(H)$.

Having thus defined the point set of $|G|$, let us choose a basis of open sets to define its topology. For every edge $u v$, declare as open all subsets of $(u, v)$ that correspond, by our fixed bijection between $(u, v)$ and $(0,1)$, to an open set in $(0,1)$. For every vertex $u$ and $\epsilon>0$, declare as open the ‘open star around $u$ of radius $\epsilon$ ‘, that is, the set of all points on edges $[u, v]$ at distance less than $\epsilon$ from $u$, measured individually for each edge in its metric inherited from $[0,1]$. Finally, for every end $\omega$ and every finite set $S \subseteq V$, there is a unique component $C(S, \omega)$ of $G-S$ that contains a ray from $\omega$. Let $\Omega(S, \omega):=\left{\omega^{\prime} \in \Omega \mid C\left(S, \omega^{\prime}\right)=C(S, \omega)\right}$. For every $\epsilon>0$, write $E_\epsilon(S, \omega)$ for the set of all inner points of $S$ $C(S, \omega)$ edges at distance less than $\epsilon$ from their endpoint in $C(S, \omega)$. Then declare as open all sets of the form
$$\hat{C}\epsilon(S, \omega):=C(S, \omega) \cup \Omega(S, \omega) \cup \dot{E}\epsilon(S, \omega) .$$

数学代写|图论作业代写Graph Theory代考|Ramsey’s original theorems

In its simplest version, Ramsey’s theorem says that, given an integer $r \geqslant 0$, every large enough graph $G$ contains either $K^r$ or $\overline{K^r}$ as an induced subgraph. At first glance, this may seem surprising: after all, we need about $(r-2) /(r-1)$ of all possible edges to force a $K^r$ subgraph in $G$ (Corollary 7.1 .3 ), but neither $G$ nor $\bar{G}$ can be expected to have more than half of all possible edges. However, as the Turán graphs illustrate well, squeezing many edges into $G$ without creating a $K^r$ imposes additional structure on $G$, which may help us find an induced $\overline{K^r}$.

So how could we go about proving Ramsey’s theorem? Let us try to build a $K^r$ or $\overline{K^r}$ in $G$ inductively, starting with an arbitrary vertex $v_1 \in V_1:=V(G)$. If $|G|$ is large, there will be a large set $V_2 \subseteq V_1 \backslash\left{v_1\right}$ of vertices that are either all adjacent to $v_1$ or all non-adjacent to $v_1$. Accordingly, we may think of $v_1$ as the first vertex of a $K^r$ or $\overline{K^r}$ whose other vertices all lie in $V_2$. Let us then choose another vertex $v_2 \in V_2$ for our $K^r$ or $\overline{K^r}$. Since $V_2$ is large, it will have a subset $V_3$, still fairly large, of vertices that are all ‘of the same type’ with respect to $v_2$ as well: either all adjacent or all non-adjacent to it. We then continue our search for vertices inside $V_3$, and so on (Fig. 9.1.1).

How long can we go on in this way? This depends on the size of our initial set $V_1$ : each set $V_i$ has at least half the size of its predecessor $V_{i-1}$, so we shall be able to complete $s$ construction steps if $G$ has order about $2^s$. As the following proof shows, the choice of $s=2 r-3$ vertices $v_i$ suffices to find among them the vertices of a $K^r$ or $\overline{K^r}$.
Theorem 9.1.1. (Ramsey 1930)
For every $r \in \mathbb{N}$ there exists an $n \in \mathbb{N}$ such that every graph of order at least $n$ contains either $K^r$ or $\overline{K^r}$ as an induced subgraph.

Proof. The assertion is trivial for $r \leqslant 1$; we assume that $r \geqslant 2$. Let $n:=2^{2 r-3}$, and let $G$ be a graph of order at least $n$. We shall define a sequence $V_1, \ldots, V_{2 r-2}$ of sets and choose vertices $v_i \in V_i$ with the following properties:
(i) $\left|V_i\right|=2^{2 r-2-i} \quad(i=1, \ldots, 2 r-2)$;

(ii) $V_i \subseteq V_{i-1} \backslash\left{v_{i-1}\right} \quad(i=2, \ldots, 2 r-2)$;
(iii) $v_{i-1}$ is adjacent either to all vertices in $V_i$ or to no vertex in $V_i$ $(i=2, \ldots, 2 r-2)$.

图论代考

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。

数学代写|微积分代写Calculus代写|MATH171

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数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

In Section 1.4, you studied functions from an algebraic point of view. In this section, you will study functions from a graphical perspective.

The graph of a function $f$ is the collection of ordered pairs $(x, f(x))$ such that $x$ is in the domain of $f$. As you study this section, remember that
$x=$ the directed distance from the $y$-axis
$y=f(x)=$ the directed distance from the $x$-axis
as shown in Figure 1.52.

Use the graph of the function $f$, shown in Figure 1.53, to find (a) the domain of $f$, (b) the function values $f(-1)$ and $f(2)$, and (c) the range of $f$.
Solution
a. The closed dot at $(-1,1)$ indicates that $x=-1$ is in the domain of $f$, whereas the open dot at $(5,2)$ indicates that $x=5$ is not in the domain. So, the domain of $f$ is all $x$ in the interval $[-1,5)$.
b. Because $(-1,1)$ is a point on the graph of $f$, it follows that $f(-1)=1$. Similarly, because $(2,-3)$ is a point on the graph of $f$, it follows that $f(2)=-3$.
c. Because the graph does not extend below $f(2)=-3$ or above $f(0)=3$, the range of $f$ is the interval $[-3,3]$.
VCHECKPOINT Now try Exercise 1.
The use of dots (open or closed) at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points.

By the definition of a function, at most one $y$-value corresponds to a given $x$-value. This means that the graph of a function cannot have two or more different points with the same $x$-coordinate, and no two points on the graph of a function can be vertically above or below each other. It follows, then, that a vertical line can intersect the graph of a function at most once. This observation provides a convenient visual test called the Vertical Line Test for functions.
Vertical Line Test for Functions
A set of points in a coordinate plane is the graph of $y$ as a function of $x$ if and only if no vertical line intersects the graph at more than one point.

数学代写|微积分代写Calculus代写|Zeros of a Function

If the graph of a function of $x$ has an $x$-intercept at $(a, 0)$, then $a$ is a zero of the function.
Zeros of a Function
The zeros of a function $f$ of $x$ are the $x$-values for which $f(x)=0$.
Example 3 Finding the Zeros of a Function
Find the zeros of each function.
a. $f(x)=3 x^2+x-10$
b. $g(x)=\sqrt{10-x^2}$
c. $h(t)=\frac{2 t-3}{t+5}$
Solution
To find the zeros of a function, set the function equal to zero and solve for the independent variable.
a.
\begin{aligned} & 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \ & (3 x-5)(x+2)=0 \ & \text { Factor. } \ & 3 x-5=0 \ & x=\frac{5}{3} \ & \text { Set } 1 \text { st factor equal to } 0 \text {. } \ & x+2=0 \ & x=-2 \ & \text { Set } 2 \text { nd factor equal to } 0 \text {. } \ & \end{aligned}
The zeros of $f$ are $x=\frac{5}{3}$ and $x=-2$. In Figure 1.55, note that the graph of $f$ has $\left(\frac{5}{3}, 0\right)$ and $(-2,0)$ as its $x$-intercepts.
b. $\sqrt{10-x^2}=0$
Set $g(x)$ equal to 0 .
\begin{aligned} 10-x^2 & =0 \ 10 & =x^2 \ \pm \sqrt{10} & =x \end{aligned}
Square each side.
Add $x^2$ to each side.
Extract square roots.
The zeros of $g$ are $x=-\sqrt{10}$ and $x=\sqrt{10}$. In Figure 1.56, note that the graph of $g$ has $(-\sqrt{10}, 0)$ and $(\sqrt{10}, 0)$ as its $x$-intercepts.

微积分代考

数学代写|微积分代写Calculus代写|Analyzing Graphs of Functions

$x=$到$y$轴的有向距离
$y=f(x)=$到$x$轴的有向距离

a.“$(-1,1)$”表示“$x=-1$”在$f$的域中，“$(5,2)$”表示“$x=5$”不在该域中。所以，$f$的定义域都是$x$在$[-1,5)$区间内。
b.因为$(-1,1)$是$f$图上的一个点，所以可知$f(-1)=1$。同样，因为$(2,-3)$是$f$图上的一个点，所以$f(2)=-3$。
c.由于图在$f(2)=-3$以下或$f(0)=3$以上不扩展，所以$f$的范围为区间$[-3,3]$。

数学代写|微积分代写Calculus代写|Zeros of a Function

$x$的函数$f$的零点是$x$ -值，$f(x)=0$。

A. $f(x)=3 x^2+x-10$
B. $g(x)=\sqrt{10-x^2}$
C. $h(t)=\frac{2 t-3}{t+5}$

a。
\begin{aligned} & 3 x^2+x-10=0 \quad \text { Set } f(x) \text { equal to } 0 . \ & (3 x-5)(x+2)=0 \ & \text { Factor. } \ & 3 x-5=0 \ & x=\frac{5}{3} \ & \text { Set } 1 \text { st factor equal to } 0 \text {. } \ & x+2=0 \ & x=-2 \ & \text { Set } 2 \text { nd factor equal to } 0 \text {. } \ & \end{aligned}
$f$的零点分别是$x=\frac{5}{3}$和$x=-2$。在图1.55中，请注意$f$的图形有$\left(\frac{5}{3}, 0\right)$和$(-2,0)$作为其$x$ -截点。
B. $\sqrt{10-x^2}=0$

\begin{aligned} 10-x^2 & =0 \ 10 & =x^2 \ \pm \sqrt{10} & =x \end{aligned}

$g$的零点分别是$x=-\sqrt{10}$和$x=\sqrt{10}$。在图1.56中，请注意$g$的图形有$(-\sqrt{10}, 0)$和$(\sqrt{10}, 0)$作为其$x$ -截点。

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。