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assignmentutor-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• 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|电动力学代写electromagnetism代考|Higher order tensors

Now we introduce a tensor product of two covectors $a$ and $b$ as $T=a b$, which acts on two vectors and yield a scalar as
$$T: x y=(a b): x y=(a \cdot x)(b \cdot y) .$$
It can be considered as a bi-linear functions of vectors, i.e., $T: x y=\Phi(x, y)$ with
\begin{aligned} &\Phi\left(c_{1} x_{1}+c_{2} x_{2}, y\right)=c_{1} \Phi\left(x_{1}, y\right)+c_{1} \Phi\left(x_{2}, y\right) \ &\Phi\left(x, c_{1} y+c_{2} y_{2}\right)=c_{1} \Phi\left(x, y_{1}\right)+c_{1} \Phi\left(x, y_{2}\right) \end{aligned}
where $c_{1}, c_{2} \in \mathbb{R}$. We call it a bi-covector.
We can define a weighted sum of bi-covectors $T=d_{1} T_{1}+d_{2} T_{2}, d_{1}, d_{2} \in \mathbb{R}$, which is not necessarily written as a tensor product of two covectors but can be written as a sum of tensor products. Especially, it can be represented with the dual basis as
$$T=\sum_{i=1}^{3} \sum_{j=1}^{3} T_{i j} \boldsymbol{n}{i} \boldsymbol{n}{j},$$
where $T_{i j}=T: e_{i} e_{j}$ is the $(i, j)$-component of $T$.
Similarly we can construct a tensor product of three covectors as $\mathcal{T}=a b c$, which acts on three vectors linearly as $\mathcal{T}: x y z$. Weighted sums of such products form a linear space, an element of which is called a tri-covector. Using a tensor product of $n$ covectors, a multi-covector or an $n$-covector is defined.

## 物理代写|电动力学代写electromagnetism代考|Anti-symmetric multi-covectors — n-forms

If a bicovector $T$ satisfies $T: y x=-T: x y$ for any vectors $x$ and $y$, then it is called antisymmetric. Anti-symmetric bicovectors form a subspace of the bicovector space. Namely, a weighted sum of anti-symmetric bicovector is anti-symmetric. It contains an anti-symmetrized tensor product, $\boldsymbol{a} \wedge \boldsymbol{b}:=\boldsymbol{a b}-\boldsymbol{b} \boldsymbol{a}$, which is called a wedge product. In terms of basis, we have
$$\boldsymbol{a} \wedge \boldsymbol{b}=\sum_{i=1}^{3} a_{i} \boldsymbol{n}{i} \wedge \sum{j=1}^{3} b_{j} \boldsymbol{n}{j}=\sum{(i, j)}\left(a_{i} b_{j}-a_{j} b_{i}\right) \boldsymbol{n}{i} \wedge \boldsymbol{n}{j},$$

where the last sum is taken for $(i, j)=(1,2),(2,3),(3,1)$. A general anti-symmetric bicovector can be written as
$$T=\sum_{(i, j)} T_{i j} \boldsymbol{n}{i} \wedge \boldsymbol{n}{j} .$$
We see that the 2-form has three independent components; $T_{12}=-T_{21}, T_{23}=-T_{32}, T_{31}=$ $-T_{13}$, and others are zero. The norm of $T$ is $|T|=(T, T)^{1 / 2}=\sum_{(i, j)} T_{i j} T_{i j}$.

If a bicovector $T$ satisfies $T: x x=0$ for any $x$, then it is anti-symmetric. It is easily seen from the relation: $0=T:(x+y)(x+y)=T: x x+T: x y+T: y x+T: y y$.

An anti-symmetric multi-covector of order $n$ are often called an $n$-form. A scalar and a covector are called a 0-form and a 1-form, respectively. The order $n$ is bounded by the dimension of the vector space, $d=3$, in our case. An $n$-form with $n>d$ vanishes due to the anti-symmetries.

Geometrical interpretations of $n$-forms are given in the articles (Misner et al. (1973); Weinreich (1998))

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Higher order tensors

$T: x y=(a b): x y=(a \backslash c d o t x)(b \backslash c d o t y)$
$\$ \$$可以认为是向量的双线性函数，即 \ T: x y=\langle P h i(x, y) \$$ 与
$\$ \begin{aligned} \& \ Phi \eft(c__1 } x_{-}{1}+c_{-}{2} x_{-}{2}, y \backslash right )=c_{-}{1} \backslash Phi \backslash left(x_ {1}, y \backslash right) +c_{-}{1} \backslash Phi \backslash left \left(x_{-}{2}, y \backslash r i g h t\right) \backslash \& Phi \left } ( x _ { 1 } , c _ { – } { 1 } y ^ { \prime } c _ { – } { 2 } y _ { – } { 2 } \backslash \text { right } ) = c _ { – } { 1 } \backslash P h i \backslash \text { left } ( x _ { 1 } y _ { – } { 1 } \backslash \text { right } ) + c _ { – } { 1 } \backslash P h i \backslash \text { eft } ( x _ { 1 } y _ { – } { 2 } \backslash \text { right } ) lend{aligned } \ \

$\$ \$$\ \$$

$\$ \$$其中最后一个总和为 \(i, j)=(1,2),(2,3),(3,1) \$$ 。般的反对称双向量可以写成
$\$ \$$T=\backslash sum_{ {(i, j)} T_{-}{i j} \backslash boldsymbol{n}{i} \backslash wedge \backslash boldsymbol{n}{j}。 \ \$$

## 有限元方法代写

assignmentutor™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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
The provided text passage is a technical discussion on tensors and multi-covectors in the context of electrodynamics. It introduces the concept of tensor products, bi-covectors, and anti-symmetric multi-covectors (n-forms), and provides mathematical definitions and properties for these constructs.
no_question

the passage does not contain any explicit mathematical questions or problems to solve. it is purely expository, providing definitions and properties of tensors and multi-covectors.
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
assignmentutor™您的专属作业导师

assignmentutor-lab™ 为您的留学生涯保驾护航 在代写电动力学electrodynamics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。我们的专家在代写电动力学electrodynamics代写方面经验极为丰富，各种代写电动力学electrodynamics相关的作业也就用不着说。

• 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|电动力学代写electromagnetism代考|Higher order tensors

Now we introduce a tensor product of two covectors $a$ and $b$ as $T=a b$, which acts on two vectors and yield a scalar as
$$T: x y=(a b): x y=(a \cdot x)(b \cdot y) .$$
It can be considered as a bi-linear functions of vectors, i.e., $T: x y=\Phi(x, y)$ with
\begin{aligned} &\Phi\left(c_{1} x_{1}+c_{2} x_{2}, y\right)=c_{1} \Phi\left(x_{1}, y\right)+c_{1} \Phi\left(x_{2}, y\right) \ &\Phi\left(x, c_{1} y+c_{2} y_{2}\right)=c_{1} \Phi\left(x, y_{1}\right)+c_{1} \Phi\left(x, y_{2}\right) \end{aligned}
where $c_{1}, c_{2} \in \mathbb{R}$. We call it a bi-covector.
We can define a weighted sum of bi-covectors $T=d_{1} T_{1}+d_{2} T_{2}, d_{1}, d_{2} \in \mathbb{R}$, which is not necessarily written as a tensor product of two covectors but can be written as a sum of tensor products. Especially, it can be represented with the dual basis as
$$T=\sum_{i=1}^{3} \sum_{j=1}^{3} T_{i j} \boldsymbol{n}{i} \boldsymbol{n}{j},$$
where $T_{i j}=T: e_{i} e_{j}$ is the $(i, j)$-component of $T$.
Similarly we can construct a tensor product of three covectors as $\mathcal{T}=a b c$, which acts on three vectors linearly as $\mathcal{T}: x y z$. Weighted sums of such products form a linear space, an element of which is called a tri-covector. Using a tensor product of $n$ covectors, a multi-covector or an $n$-covector is defined.

## 物理代写|电动力学代写electromagnetism代考|Anti-symmetric multi-covectors — n-forms

If a bicovector $T$ satisfies $T: y x=-T: x y$ for any vectors $x$ and $y$, then it is called antisymmetric. Anti-symmetric bicovectors form a subspace of the bicovector space. Namely, a weighted sum of anti-symmetric bicovector is anti-symmetric. It contains an anti-symmetrized tensor product, $\boldsymbol{a} \wedge \boldsymbol{b}:=\boldsymbol{a b}-\boldsymbol{b} \boldsymbol{a}$, which is called a wedge product. In terms of basis, we have
$$\boldsymbol{a} \wedge \boldsymbol{b}=\sum_{i=1}^{3} a_{i} \boldsymbol{n}{i} \wedge \sum{j=1}^{3} b_{j} \boldsymbol{n}{j}=\sum{(i, j)}\left(a_{i} b_{j}-a_{j} b_{i}\right) \boldsymbol{n}{i} \wedge \boldsymbol{n}{j},$$

where the last sum is taken for $(i, j)=(1,2),(2,3),(3,1)$. A general anti-symmetric bicovector can be written as
$$T=\sum_{(i, j)} T_{i j} \boldsymbol{n}{i} \wedge \boldsymbol{n}{j} .$$
We see that the 2-form has three independent components; $T_{12}=-T_{21}, T_{23}=-T_{32}, T_{31}=$ $-T_{13}$, and others are zero. The norm of $T$ is $|T|=(T, T)^{1 / 2}=\sum_{(i, j)} T_{i j} T_{i j}$.

If a bicovector $T$ satisfies $T: x x=0$ for any $x$, then it is anti-symmetric. It is easily seen from the relation: $0=T:(x+y)(x+y)=T: x x+T: x y+T: y x+T: y y$.

An anti-symmetric multi-covector of order $n$ are often called an $n$-form. A scalar and a covector are called a 0-form and a 1-form, respectively. The order $n$ is bounded by the dimension of the vector space, $d=3$, in our case. An $n$-form with $n>d$ vanishes due to the anti-symmetries.

Geometrical interpretations of $n$-forms are given in the articles (Misner et al. (1973); Weinreich (1998))

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Higher order tensors

$T: x y=(a b): x y=(a \backslash c d o t x)(b \backslash c d o t y)$
$\$ \$$可以认为是向量的双线性函数，即 \ T: x y=\langle P h i(x, y) \$$ 与
$\$ \begin{aligned} \& \ Phi \eft(c__1 } x_{-}{1}+c_{-}{2} x_{-}{2}, y \backslash right )=c_{-}{1} \backslash Phi \backslash left(x_ {1}, y \backslash right) +c_{-}{1} \backslash Phi \backslash left \left(x_{-}{2}, y \backslash r i g h t\right) \backslash \& Phi \left } ( x _ { 1 } , c _ { – } { 1 } y ^ { \prime } c _ { – } { 2 } y _ { – } { 2 } \backslash \text { right } ) = c _ { – } { 1 } \backslash P h i \backslash \text { left } ( x _ { 1 } y _ { – } { 1 } \backslash \text { right } ) + c _ { – } { 1 } \backslash P h i \backslash \text { eft } ( x _ { 1 } y _ { – } { 2 } \backslash \text { right } ) lend{aligned } \ \

$\$ \$$\ \$$

$\$ \$$其中最后一个总和为 \(i, j)=(1,2),(2,3),(3,1) \$$ 。般的反对称双向量可以写成
$\$ \$$T=\backslash sum_{ {(i, j)} T_{-}{i j} \backslash boldsymbol{n}{i} \backslash wedge \backslash boldsymbol{n}{j}。 \ \$$

## 有限元方法代写

assignmentutor™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。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师
user
The provided text passage is a technical discussion on tensors and multi-covectors in the context of electrodynamics. It introduces the concept of tensor products, bi-covectors, and anti-symmetric multi-covectors (n-forms), and provides mathematical definitions and properties for these constructs.

TYPE: NO_QUESTION

The passage does not contain any explicit mathematical questions or problems to solve. It is purely expository, providing definitions and properties of tensors and multi-covectors.