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## 数学代写|变分法代写Calculus of Variations代考|Summation Convention, Kronecker Delta

A space vector $\boldsymbol{a}$ can be represented as
$$\boldsymbol{a}=a_1 \boldsymbol{e}1+a_2 \boldsymbol{e}_2+a_3 \boldsymbol{e}_3=\sum{i=1}^3 a_i \boldsymbol{e}_i$$
In order to make Eq. (1.6.1) express more concise, it can make such a convention: In a certain term of an expression, when a certain index (superscript or subscript) appears twice, which means the index summation within scope, at the same time the summation symbol is omitted, such a convention is called the Einstein summation convention. In a certain term of an expression, the repeated index is called a dummy index. Then Eq. (1.6.1) can be abbreviated to
$$\boldsymbol{a}=a_i \boldsymbol{e}_i$$
The dummy index only indicates summation, it doesn’t matter what symbol is used to express the dummy index, but the symbol that has a specific meaning should be avoided using.

In $n$-dimensional space, any group of $n$ linearly independent vectors are called their a base or basis, every vector in the base is called the base vector. In the rectangular coordinate system, the base composed of three mutually perpendicular base vectors with unit length is called the rectangular Cartesian base (basis), it is called the Cartesian base (basis) for short.

In three-dimensional rectangular coordinate system, the base vectors $\boldsymbol{e}1, \boldsymbol{e}_2$ and $e_3$ are perpendicular to each other and the modulus of each base vector is unity, that is $$\boldsymbol{e}_1 \cdot \boldsymbol{e}_1=1, \boldsymbol{e}_2 \cdot \boldsymbol{e}_2=1, \boldsymbol{e}_3 \cdot \boldsymbol{e}_3=1, \boldsymbol{e}_1 \cdot \boldsymbol{e}_2=0, \boldsymbol{e}_2 \cdot \boldsymbol{e}_3=0, \boldsymbol{e}_3 \cdot \boldsymbol{e}_1=0$$ These six relations can be concisely expressed by the relation $$\boldsymbol{e}_i \cdot \boldsymbol{e}_j=\delta{i j}$$
where
$$\delta_{i j}= \begin{cases}1 & i=j \ 0 & i \neq j\end{cases}$$

## 数学代写|变分法代写Calculus of Variations代考|Rotation Transformations of Rectangle Coordinates

In the rectangular coordinate system $O x y z$, a vector $a$ can be represented with its three components $a_x, a_y$ and $a_z$. Due to the coordinate system is artificially chosen, of course another coordinate system $O x^{\prime} y^{\prime} z^{\prime}$ can also be chosen, the three components of $\boldsymbol{a}$ become $a_{x^{\prime}}, a_{y^{\prime}}$ and $a_{z^{\prime}}$. The same vector $\boldsymbol{a}$, in different coordinate systems can be represented in different components.

Let $O x_1 x_2 x_3$ and $O x_1^{\prime} x_2^{\prime} x_3^{\prime}$ be an old and a new right-handed rectangular coordinate systems respectively. $\boldsymbol{e}1, \boldsymbol{e}_2, \boldsymbol{e}_3$ and $\boldsymbol{e}_1^{\prime}, \boldsymbol{e}_2^{\prime}, \boldsymbol{e}_3^{\prime}$ are the unit vectors on the axes in the two coordinate systems respectively, then there is $$\boldsymbol{e}_i \cdot \boldsymbol{e}_j=\boldsymbol{e}_i^{\prime} \cdot \boldsymbol{e}_j^{\prime}=\delta{i j}$$
There are the following relationships between the unit vectors in the old and new coordinates
$$\left{\begin{array}{l} \boldsymbol{e}1^{\prime}=\alpha{11} \boldsymbol{e}1+\alpha{12} \boldsymbol{e}2+\alpha{13} \boldsymbol{e}3 \ \boldsymbol{e}_3^{\prime}=\alpha{21} \boldsymbol{e}1+\alpha{22} \boldsymbol{e}2+\alpha{23} \boldsymbol{e}3 \ \boldsymbol{e}_3^{\prime}=\alpha{31} \boldsymbol{e}1+\alpha{32} \boldsymbol{e}2+\alpha{33} \boldsymbol{e}3 \end{array}\right.$$ where, $\alpha{i j}=\boldsymbol{e}i^{\prime} \cdot \boldsymbol{e}_j$ is the cosine of included angle of the different axes in the two coordinate systems, namely the direction cosine, it is called the coefficient of transformation, The first index expresses the new coordinates, The second index expresses the old coordinates. Making use of Einstein summation convention, Eq. (1.7.2) can be written as $$\boldsymbol{e}_i^{\prime}=\alpha{i j} \boldsymbol{e}j, \boldsymbol{e}_i=\alpha{j i} \boldsymbol{e}_j^{\prime}$$
Let the three components of the vector $\boldsymbol{a}$ in the old coordinate system be $a_1, a_2$, $a_3$, the three components in the new coordinate system are $a_1^{\prime}, a_2^{\prime}, a_3^{\prime}$. In the old and the new coordinate system, the vector $\boldsymbol{a}$ can be represented as $$\boldsymbol{a}=\left(\boldsymbol{a} \cdot \boldsymbol{e}j\right) \boldsymbol{e}_j=a_j \boldsymbol{e}_j$$ $$\boldsymbol{a}=\left(\boldsymbol{a} \cdot \boldsymbol{e}_i^{\prime}\right) \boldsymbol{e}_i^{\prime}=a_i^{\prime} \boldsymbol{e}_i^{\prime}$$ Substituting Eq. (1.7.4) into Eq. (1.7.5), dot-multiplying both sides of Eq. (1.7.5) by $\boldsymbol{e}_i^{\prime}$, then there is the following relations between the components in the new coordinate system and the ones of the old coordinate system $$a_i^{\prime}=a_i^{\prime} \boldsymbol{e}_i^{\prime} \cdot \boldsymbol{e}_i^{\prime}=\boldsymbol{e}_i^{\prime} \cdot \boldsymbol{e}_j a_j=\alpha{i j} a_j$$
where, $i$ is a free index. Expanding Eq. (1.7.6), there is
$$\left{\begin{array}{l} a_1^{\prime}=\boldsymbol{a} \cdot \boldsymbol{e}1^{\prime}=\left(a_j \boldsymbol{e}_j\right) \cdot \boldsymbol{e}_1^{\prime}=\alpha{11} a_1+\alpha_{12} a_2+\alpha_{13} a_3 \ a_2^{\prime}=\boldsymbol{a} \cdot \boldsymbol{e}2^{\prime}=\left(a_j \boldsymbol{e}_j\right) \cdot \boldsymbol{e}_2^{\prime}=\alpha{21} a_1+\alpha_{22} a_2+\alpha_{23} a_3 \ a_3^{\prime}=\boldsymbol{a} \cdot \boldsymbol{e}3^{\prime}=\left(a_j \boldsymbol{e}_j\right) \cdot \boldsymbol{e}_3^{\prime}=\alpha{31} a_1+\alpha_{32} a_2+\alpha_{33} a_3 \end{array}\right.$$

# 变分法代考

## 数学代写|变分法代写Calculus of Variations代考|Summation Convention, Kronecker Delta

$$\boldsymbol{a}=a_1 \boldsymbol{e} 1+a_2 \boldsymbol{e}2+a_3 \boldsymbol{e}_3=\sum i=1^3 a_i \boldsymbol{e}_i$$ 为了使方程式。(1.6.1)表达式更简洁，可以做这样的约定：在表达式的某个术语中，当某个索引 (上标或下标) 出现两次时，表示范围内的索引求和，同时求和符 号被省略，这样的约定称为爱因斯坦求和约定。在表达式的某个术语中，重复的索引称为哑索引。然后等式。(1.6.1)可简写为 $$\boldsymbol{a}=a_i \boldsymbol{e}_i$$ 哑索引只表示求和，用什么符号表示哑索引无关紧要，但应避免使用具有特定含义的符号。 在 $n$ 维空间，任何一组 $n$ 线性独立的向量称为基或基，基中的每个向量称为基向量。在直角坐标系中，由三个相互垂直的单位长度基向量组成的底称为直角笛卡尔 底 (basis)，简称笛卡尔底 (basis) 。 在三维直角坐标系中，基向量 $e 1, e_2$ 和 $e_3$ 相互垂直且每个基向量的模为一，即 $$\boldsymbol{e}_1 \cdot \boldsymbol{e}_1=1, \boldsymbol{e}_2 \cdot \boldsymbol{e}_2=1, \boldsymbol{e}_3 \cdot \boldsymbol{e}_3=1, \boldsymbol{e}_1 \cdot \boldsymbol{e}_2=0, \boldsymbol{e}_2 \cdot \boldsymbol{e}_3=0, \boldsymbol{e}_3 \cdot \boldsymbol{e}_1=0$$ 这六种关系可以简明地表达为 $$\boldsymbol{e}_i \cdot \boldsymbol{e}_j=\delta i j$$ 在哪里 $$\delta{i j}=\left{\begin{array}{l} 1 \quad i=j 0 \quad i \neq j \end{array}\right.$$

## 数学代写|变分法代写Calculus of Variations代考|Rotation Transformations of Rectangle Coordinates

$$\boldsymbol{e}_i \cdot \boldsymbol{e}_j=\boldsymbol{e}_i^{\prime} \cdot \boldsymbol{e}_j^{\prime}=\delta i j$$

Neft{中的单位向量有如下关系
$$e 1^{\prime}=\alpha 11 e 1+\alpha 12 e 2+\alpha 13 e 3 e_3^{\prime}=\alpha 21 e 1+\alpha 22 e 2+\alpha 23 e 3 e_3^{\prime}=\alpha 31 e 1+\alpha 32 e 2+\alpha 33 e 3$$

where, $\$ \alpha i j=e^{\prime} \cdot e_j \$i s t h e c o s i n e o f i n c l u d e d a n g l e o f t h e d i f f e r e n t a x e s i n t h e t w o c o o r d i n a t e s y s t e m s$, namelythedirectioncosine, itiscalledthecoeff
SubstitutingEq. (1.7.4)intoEq. (1.7.5), dot – multiplyingbothsidesofEq. (1.7.5)by\$e$\$$\, thenthereisthefollowingrelationsbetweenthecomponentsir where, \ i \$$ isafreeindex. ExpandingEq. (1.7.6), thereis

## 有限元方法代写

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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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