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# 物理代写|电动力学代考Electrodynamics代写|Spherical Coordinates

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## 物理代写|电动力学代考Electrodynamics代写|Spherical Coordinates

Figure 1.2(b) defines spherical coordinates $(r, \theta, \phi)$. Our notation for the components and unit vectors in this system is
$$\mathbf{V}=V_r \hat{\mathbf{r}}+V_\theta \hat{\boldsymbol{\theta}}+V_\phi \hat{\boldsymbol{\phi}}$$
The transformation to Cartesian coordinates is
$$x=r \sin \theta \cos \phi \quad y=r \sin \theta \sin \phi \quad z=r \cos \theta .$$
The volume element in spherical coordinates is $d^3 r=r^2 \sin \theta d r d \theta d \phi$. The unit vectors are related by
$$\begin{array}{cc} \hat{\mathbf{r}}=\hat{\mathbf{x}} \sin \theta \cos \phi+\hat{\mathbf{y}} \sin \theta \sin \phi+\hat{\mathbf{z}} \cos \theta & \hat{\mathbf{x}}=\hat{\mathbf{r}} \sin \theta \cos \phi+\hat{\boldsymbol{\theta}} \cos \theta \cos \phi-\hat{\boldsymbol{\phi}} \sin \phi \ \hat{\boldsymbol{\theta}}=\hat{\mathbf{x}} \cos \theta \cos \phi+\hat{\mathbf{y}} \cos \theta \sin \phi-\hat{\mathbf{z}} \sin \theta & \hat{\mathbf{y}}=\hat{\mathbf{r}} \sin \theta \sin \phi+\hat{\boldsymbol{\theta}} \cos \theta \sin \phi+\hat{\boldsymbol{\phi}} \cos \phi \ \hat{\boldsymbol{\phi}}=-\hat{\mathbf{x}} \sin \phi+\hat{\mathbf{y}} \cos \phi & \hat{\mathbf{z}}=\hat{\mathbf{r}} \cos \theta-\hat{\boldsymbol{\theta}} \sin \theta . \end{array}$$
The gradient operator in spherical coordinates is
$$\nabla=\hat{\mathbf{r}} \frac{\partial}{\partial r}+\frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial}{\partial \theta}+\frac{\hat{\boldsymbol{\phi}}}{r \sin \theta} \frac{\partial}{\partial \phi} .$$
The divergence, curl, and Laplacian operations are, respectively,
\begin{aligned} \nabla \cdot \mathbf{V}= & \frac{1}{r^2} \frac{\partial\left(r^2 V_r\right)}{\partial r}+\frac{1}{r \sin \theta} \frac{\partial\left(\sin \theta V_\theta\right)}{\partial \theta}+\frac{1}{r \sin \theta} \frac{\partial V_\phi}{\partial \phi} \ \nabla \times \mathbf{V}= & \frac{1}{r \sin \theta}\left[\frac{\partial\left(\sin \theta V_\phi\right)}{\partial \theta}-\frac{\partial V_\theta}{\partial \phi}\right] \hat{\mathbf{r}} \ & +\frac{1}{r}\left[\frac{1}{\sin \theta} \frac{\partial V_r}{\partial \phi}-\frac{\partial\left(r V_\phi\right)}{\partial r}\right] \hat{\boldsymbol{\theta}}+\frac{1}{r}\left[\frac{\partial\left(r V_\theta\right)}{\partial r}-\frac{\partial V_r}{\partial \theta}\right] \hat{\boldsymbol{\phi}} \ \nabla^2 A= & \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial A}{\partial r}\right)+\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial A}{\partial \theta}\right)+\frac{1}{r^2 \sin ^2 \theta} \frac{\partial^2 A}{\partial \phi^2} . \end{aligned}

## 物理代写|电动力学代考Electrodynamics代写|The Einstein Summation Convention

Einstein (1916) introduced the following convention. An index which appears exactly twice in a mathematical expression is implicitly summed over all possible values for that index. The range of this dummy index must be clear from context and the index cannot be used elsewhere in the same expression for another purpose. In this book, the range for a roman index like $i$ is from 1 to 3 , indicating a sum over the Cartesian indices $x, y$, and $z$. Thus, $\mathbf{V}$ in (1.6) and its dot product with another vector F are written
$$\mathbf{V}=\sum_{k=1}^3 V_k \hat{\mathbf{e}}k \equiv V_k \hat{\mathbf{e}}_k \quad \mathbf{V} \cdot \mathbf{F}=\sum{k=1}^3 V_k F_k \equiv V_k F_k .$$
In a Cartesian basis, the gradient of a scalar $\varphi$ and the divergence of a vector $\mathbf{D}$ can be variously written
\begin{aligned} & \nabla \varphi=\hat{\mathbf{e}}k \nabla_k \varphi=\hat{\mathbf{e}}_k \partial_k \varphi=\hat{\mathbf{e}}_k \frac{\partial \varphi}{\partial r_k} \ & \nabla \cdot \mathbf{D}=\nabla_k D_k=\partial_k D_k=\frac{\partial D_k}{\partial r_k} . \end{aligned} If an $N \times N$ matrix $\mathbf{C}$ is the product of an $N \times M$ matrix $\mathbf{A}$ and an $M \times N$ matrix $\mathbf{B}$, $$C{i k}=\sum_{j=1}^M A_{i j} B_{j k}=A_{i j} B_{j k} .$$

# 电动力学代写

## 物理代写|电动力学代考Electrodynamics代写|Spherical Coordinates

$$\mathbf{V}=V_r \hat{\mathbf{r}}+V_\theta \hat{\boldsymbol{\theta}}+V_\phi \hat{\boldsymbol{\phi}}$$

$$x=r \sin \theta \cos \phi \quad y=r \sin \theta \sin \phi \quad z=r \cos \theta .$$

$$\begin{array}{cc} \hat{\mathbf{r}}=\hat{\mathbf{x}} \sin \theta \cos \phi+\hat{\mathbf{y}} \sin \theta \sin \phi+\hat{\mathbf{z}} \cos \theta & \hat{\mathbf{x}}=\hat{\mathbf{r}} \sin \theta \cos \phi+\hat{\boldsymbol{\theta}} \cos \theta \cos \phi-\hat{\boldsymbol{\phi}} \sin \phi \ \hat{\boldsymbol{\theta}}=\hat{\mathbf{x}} \cos \theta \cos \phi+\hat{\mathbf{y}} \cos \theta \sin \phi-\hat{\mathbf{z}} \sin \theta & \hat{\mathbf{y}}=\hat{\mathbf{r}} \sin \theta \sin \phi+\hat{\boldsymbol{\theta}} \cos \theta \sin \phi+\hat{\boldsymbol{\phi}} \cos \phi \ \hat{\boldsymbol{\phi}}=-\hat{\mathbf{x}} \sin \phi+\hat{\mathbf{y}} \cos \phi & \hat{\mathbf{z}}=\hat{\mathbf{r}} \cos \theta-\hat{\boldsymbol{\theta}} \sin \theta . \end{array}$$

$$\nabla=\hat{\mathbf{r}} \frac{\partial}{\partial r}+\frac{\hat{\boldsymbol{\theta}}}{r} \frac{\partial}{\partial \theta}+\frac{\hat{\boldsymbol{\phi}}}{r \sin \theta} \frac{\partial}{\partial \phi} .$$

\begin{aligned} \nabla \cdot \mathbf{V}= & \frac{1}{r^2} \frac{\partial\left(r^2 V_r\right)}{\partial r}+\frac{1}{r \sin \theta} \frac{\partial\left(\sin \theta V_\theta\right)}{\partial \theta}+\frac{1}{r \sin \theta} \frac{\partial V_\phi}{\partial \phi} \ \nabla \times \mathbf{V}= & \frac{1}{r \sin \theta}\left[\frac{\partial\left(\sin \theta V_\phi\right)}{\partial \theta}-\frac{\partial V_\theta}{\partial \phi}\right] \hat{\mathbf{r}} \ & +\frac{1}{r}\left[\frac{1}{\sin \theta} \frac{\partial V_r}{\partial \phi}-\frac{\partial\left(r V_\phi\right)}{\partial r}\right] \hat{\boldsymbol{\theta}}+\frac{1}{r}\left[\frac{\partial\left(r V_\theta\right)}{\partial r}-\frac{\partial V_r}{\partial \theta}\right] \hat{\boldsymbol{\phi}} \ \nabla^2 A= & \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial A}{\partial r}\right)+\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial A}{\partial \theta}\right)+\frac{1}{r^2 \sin ^2 \theta} \frac{\partial^2 A}{\partial \phi^2} . \end{aligned}

## 物理代写|电动力学代考Electrodynamics代写|The Einstein Summation Convention

$$\mathbf{V}=\sum_{k=1}^3 V_k \hat{\mathbf{e}}k \equiv V_k \hat{\mathbf{e}}k \quad \mathbf{V} \cdot \mathbf{F}=\sum{k=1}^3 V_k F_k \equiv V_k F_k .$$ 在笛卡尔基中，标量的梯度$\varphi$和矢量的散度$\mathbf{D}$可以写成不同的形式 \begin{aligned} & \nabla \varphi=\hat{\mathbf{e}}k \nabla_k \varphi=\hat{\mathbf{e}}_k \partial_k \varphi=\hat{\mathbf{e}}_k \frac{\partial \varphi}{\partial r_k} \ & \nabla \cdot \mathbf{D}=\nabla_k D_k=\partial_k D_k=\frac{\partial D_k}{\partial r_k} . \end{aligned}如果$N \times N$矩阵$\mathbf{C}$是$N \times M$矩阵$\mathbf{A}$和$M \times N$矩阵$\mathbf{B}$的乘积， $$C{i k}=\sum{j=1}^M A_{i j} B_{j k}=A_{i j} B_{j k} .$$

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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。