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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• 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|Field quantities as n-forms

Field quantities in electromagnetism can be naturally represented as differential forms (Burke (1985); Deschamps (1981); Flanders (1989); Frankel (2004); Hehl \& Obukhov (2003)). A good example of 2-form is the current density. Let us consider a distribution of current that flows through a parallelogram spanned by two tangential vectors $x$ and $y$ at $r$. The current $I(x, y)$ is bilinearly dependent on $x$ and $y$. The antisymmetric relation $I(y, x)=-I(x, y)$ can understood naturally considering the orientation of parallelograms with respect to the current flow. Thus the current density can be represented by a 2-form $J$ as
$$J: x y=I(x, y) \stackrel{\mathfrak{s}}{\sim} \mathrm{A}, \quad J=\sum_{(i, j)} J_{i j} n_{i} \wedge n_{j} \stackrel{\mathfrak{s}}{\sim} \mathrm{A} / \mathrm{m}^{2} .$$
The charge density can be represented by a 3 -form $\mathcal{R}$. The charge $Q$ contained in a parallelepipedon spanned by three tangential vectors $x, y$, and $z$ :
$$\mathcal{R}: x y z=Q(x, y, z) \stackrel{\mathrm{sL}}{\sim} \mathrm{C}, \quad \mathcal{R}=R_{123} n_{1} \wedge n_{2} \wedge n_{3} \stackrel{\mathrm{st}}{\sim} \mathrm{C} / \mathrm{m}^{3} .$$
Thus electromagnetic field quantities are represented as $n$-forms $(n=0,1,2,3)$ as shown in Table 1, while in the conventional formalism they are classified into two categories, scalars and vectors, according to the number of components. We notice that a quantity that is represented $n$-form contains physical dimension with $\mathrm{m}^{-n}$ in SI. An $n$-form takes $n$ tangential vectors, each of which has dimension of length and is measured in $\mathrm{m}$ (meters).

In this article, 1-forms are represented by bold-face letters, 2-forms sans-serif letters, and 3-forms calligraphic letters as shown in Table $1 .$

## 物理代写|电动力学代写electromagnetism代考|The Maxwell equations in the differential forms

With differential forms, we can rewrite the Maxwell equations and the constitutive relations as,
\begin{aligned} &\nabla \wedge B=0, \quad \nabla \wedge E+\frac{\partial B}{\partial t}=0, \ &\nabla \wedge D=\mathcal{R}, \quad \nabla \wedge H-\frac{\partial D}{\partial t}=J, \ &D=\varepsilon_{0} \mathcal{E} \cdot E+P, \quad H=\frac{1}{2} \mu_{0}^{-1} \mathcal{E}: B-M \end{aligned}
In the formalism of differential forms, the spatial derivative $\nabla \wedge_{ப}$ is simply denoted as $d_{ப}$. Together with the Hodge operator “”, Eq. $(30)$ is written in simpler forms; \begin{aligned} &\mathrm{d} B=0, \quad \mathrm{~d} E+\partial_{t} B=0, \ &\mathrm{~d} D=\mathcal{R}, \quad \mathrm{d} H-\partial_{t} D=J, \ &D=\varepsilon_{0}( E)+P, \quad H=\mu_{0}^{-1}(* B)-M, \end{aligned}
where $\partial_{t}=\partial / \partial t$.
In Fig. 1, we show a diagram corresponding Eq. (31) and related equations (Deschamps (1981)). The field quantities are arranged according to their tensor order. The exterior derivative ” $\mathrm{d}$ ” connects a pair of quantities by increasing the tensor order by one, while time derivative $\partial_{t}$ conserves the tensor order. $E(B)$ is related to $D(H)$ with the Hodge star operator and the constant $\varepsilon_{0}\left(\mu_{0}\right)$. The definitions of potentials and the charge conservation law
$$E=-\mathrm{d} \phi-\partial_{t} A, \quad B=\mathrm{d} A, \quad \mathrm{~d} J+\partial_{t} \mathcal{R}=0$$
are also shown in Fig. 1. We can see a well-organized, perfect structure. We will see the relativistic version later (Fig. 2).

# 电动力学代考

## 物理代写|电动力学代写electromagnetism代考|Field quantities as n-forms

$$J: x y=I(x, y) \stackrel{5}{\sim} \mathrm{A}, \quad J=\sum_{(i, j)} J_{i j} n_{i} \wedge n_{j} \stackrel{5}{\sim} \mathrm{A} / \mathrm{m}^{2} .$$

$$\mathcal{R}: x y z=Q(x, y, z) \stackrel{\text { sL }}{\sim} \mathrm{C}, \quad \mathcal{R}=R_{123} n_{1} \wedge n_{2} \wedge n_{3} \stackrel{\text { st }}{\sim} \mathrm{C} / \mathrm{m}^{3} .$$

## 物理代写|电动力学代写electromagnetism代考|The Maxwell equations in the differential forms

$$\nabla \wedge B=0, \quad \nabla \wedge E+\frac{\partial B}{\partial t}=0, \quad \nabla \wedge D=\mathcal{R}, \quad \nabla \wedge H-\frac{\partial D}{\partial t}=J, D=\varepsilon_{0} \mathcal{E} \cdot E+P, \quad H=\frac{1}{2} \mu_{0}^{-1} \mathcal{E}: B-M$$

$$\mathrm{d} B=0, \quad \mathrm{~d} E+\partial_{t} B=0, \quad \mathrm{~d} D=\mathcal{R}, \quad \mathrm{d} H-\partial_{t} D=J, D=\varepsilon_{0}(E)+P, \quad H=\mu_{0}^{-1}(* B)-M,$$

$$E=-\mathrm{d} \phi-\partial_{t} A, \quad B=\mathrm{d} A, \quad \mathrm{~d} J+\partial_{t} \mathcal{R}=0$$

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## MATLAB代写

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