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<?xml version="1.0" encoding="utf-8"?>
<search>
<entry>
<title>Field Theory Calculation of Orthogonal Curve Coordinate System</title>
<url>/2022/04/09/field-theory-calculation/</url>
<content><![CDATA[<h1 id="正交曲线坐标系的场论计算">正交曲线坐标系的场论计算</h1>
<blockquote>
<p>参考：张筑生1990数学分析新讲1_北京大学</p>
</blockquote>
<h2 id="直角坐标系">直角坐标系</h2>
<p><span class="math display">\[\nabla=\vec{i}\frac{\partial}{\partial x}+\vec{j}\frac{\partial}{\partial y}+\vec{k}\frac{\partial}{\partial z}\]</span></p>
<h2 id="曲线坐标系">曲线坐标系</h2>
<p><span class="math display">\[\vec{r}=\vec{r}(x,y,z) \Rightarrow \vec{r}=\vec{r}(q_1,q_2,q_3)\]</span> <span class="math display">\[d\vec{r}=\frac{\partial\vec{r}}{\partial q_1}dq_1+\frac{\partial\vec{r}}{\partial q_2}dq_2+\frac{\partial\vec{r}}{\partial q_3}dq_3\]</span> 各坐标切向量 <span class="math display">\[\frac{\partial\vec{r}}{\partial q_1},\qquad \frac{\partial\vec{r}}{\partial q_2},\qquad \frac{\partial\vec{r}}{\partial q_3}\]</span> Lam<span class="math inline">\(\acute{\rm{e}}\)</span>系数 <span class="math display">\[h_i=\Big\|\frac{\partial\vec{r}}{\partial q_i}\Big\|=\sqrt{\Big(\frac{\partial x}{\partial q_i}\Big)^2+(\frac{\partial y}{\partial q_i}\Big)^2+(\frac{\partial z}{\partial q_i}\Big)^2},\qquad i=1,2,3\]</span> 正交曲线坐标系：柱坐标、球坐标，规范正交基底 <span class="math display">\[\vec{e}_i=\frac{1}{h_i}\frac{\partial\vec{r}}{\partial q_i},\qquad i=1,2,3\]</span> <span class="math display">\[\vec{e}_i\cdot\vec{e}_j=\delta_{ij}\]</span></p>
<h3 id="梯度">梯度</h3>
<p><span class="math display">\[d\vec{r}=h_1dq_1\vec{e}_1+h_2dq_2\vec{e}_2+h_3dq_3\vec{e}_3\]</span> <span class="math display">\[\nabla u=\lambda_1\vec{e}_1+\lambda_2\vec{e}_2+\lambda_3\vec{e}_3\]</span> <span class="math display">\[\nabla u\cdot d\vec{r}=\lambda_1h_1dq_1+\lambda_2h_2dq_2+\lambda_3h_3dq_3\]</span> <span class="math display">\[\nabla u\cdot d\vec{r}=du=\frac{\partial u}{\partial q_1}dq_1+\frac{\partial u}{\partial q_2}dq_2+\frac{\partial u}{\partial q_3}dq_3\]</span> <span class="math display">\[\Rightarrow\lambda_i=\frac{1}{h_i}\frac{\partial u}{\partial q_i},\qquad i=1,2,3\]</span> <span class="math display">\[\Rightarrow\nabla u=\frac{1}{h_1}\frac{\partial u}{\partial q_1}\vec{e}_1+\frac{1}{h_2}\frac{\partial u}{\partial q_2}\vec{e}_2+\frac{1}{h_3}\frac{\partial u}{\partial q_3}\vec{e}_3\]</span></p>
<h3 id="散度">散度</h3>
<p>由梯度公式有 <span class="math display">\[\nabla q_i=\frac{1}{h_i}\vec{e}_i,\qquad i=1,2,3\]</span> <span class="math display">\[\begin{aligned}
\nabla\cdot\Big(\frac{\vec{e}_1}{h_2h_3}\Big)&=\nabla\cdot\Big(\frac{\vec{e}_2}{h_2}\times\frac{\vec{e}_3}{h_3}\Big)=\nabla\cdot(\nabla q_2\times\nabla q_3)\\
&=(\nabla\times\nabla q_2)\cdot\nabla q_3-(\nabla\times\nabla q_3)\cdot\nabla q_2=0
\end{aligned}\]</span> <span class="math display">\[\Rightarrow\nabla\cdot\Big(\frac{\vec{e}_2}{h_3h_1}\Big)=0,\qquad\nabla\cdot\Big(\frac{\vec{e}_3}{h_1h_2}\Big)=0\]</span> <span class="math display">\[\vec{u}=u_1\vec{e}_1+u_2\vec{e}_2+u_3\vec{e}_3=u_1h_2h_3\Big(\frac{\vec{e}_1}{h_2h_3}\Big)+u_2h_3h_1\Big(\frac{\vec{e}_2}{h_3h_1}\Big)+u_3h_1h_2\Big(\frac{\vec{e}_3}{h_1h_2}\Big)\]</span> <span class="math display">\[\begin{aligned}
\nabla\cdot\vec{u}&=\nabla(u_1h_2h_3)\cdot\Big(\frac{\vec{e}_1}{h_2h_3}\Big)+\nabla(u_2h_1h_1)\cdot\Big(\frac{\vec{e}_2}{h_3h_1}\Big)+\nabla(u_3h_1h_2)\cdot\Big(\frac{\vec{e}_3}{h_1h_2}\Big)\\
&=\frac{1}{h_1h_2h_3}\Big[\frac{\partial}{\partial q_1}(u_1h_2h_3)+\frac{\partial}{\partial q_2}(u_2h_3h_1)+\frac{\partial}{\partial q_3}(u_3h_1h_2)\Big]
\end{aligned}\]</span></p>
<h3 id="旋度">旋度</h3>
<p><span class="math display">\[\nabla\times\Big(\frac{\vec{e}_i}{h_i}\Big)=\nabla\times\nabla q_i=0,\qquad i=1,2,3 \]</span> <span class="math display">\[\vec{u}=\sum_{i=1}^3u_i\vec{e}_i=\sum_{i=1}^3u_ih_i\Big(\frac{\vec{e}_i}{h_i}\Big) \]</span> <span class="math display">\[\nabla\times\vec{u}=\sum_{i=1}^3\nabla(u_ih_i)\times\Big(\frac{\vec{e}_i}{h_i}\Big)=\frac{1}{h_1h_2h_3}\begin{vmatrix}
h_1\vec{e}_1 &h_2\vec{e}_2 &h_3\vec{e}_3\\
\frac{\partial}{\partial q_1} &\frac{\partial}{\partial q_2} &\frac{\partial}{\partial q_3}\\
h_1u_1 &h_2u_2 &h_3u_3
\end{vmatrix} \]</span></p>
<h3 id="拉普拉斯算子">拉普拉斯算子</h3>
<p>直角坐标系 <span class="math display">\[\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2},\qquad \Delta u=\nabla\cdot(\nabla u) \]</span> 正交曲线坐标系 <span class="math display">\[\begin{aligned}
\Delta u&=\frac{1}{h_1h_2h_3}\Big[\frac{\partial}{\partial q_1}\Big(\frac{h_2h_3}{h_1}\frac{\partial u}{\partial q_1}\Big)+\frac{\partial}{\partial q_2}\Big(\frac{h_3h_1}{h_2}\frac{\partial u}{\partial q_2}\Big)+\frac{\partial}{\partial q_3}\Big(\frac{h_1h_2}{h_3}\frac{\partial u}{\partial q_3}\Big)\Big] \\
&=\frac{1}{h_1h_2h_3}\sum_{i=1}^3\frac{\partial}{\partial q_i}\Big(\frac{h_1h_2h_3}{h_i^2}\frac{\partial u}{\partial q_i}\Big)
\end{aligned}\]</span></p>
<h3 id="柱坐标">柱坐标</h3>
<p><span class="math display">\[\vec{r}(x,y,z)\Rightarrow\vec{r}(r,\theta,z)\qquad\left\{\begin{aligned}
x&=r\cos\theta\\
y&=r\sin\theta\\
z&=z
\end{aligned}\right.\]</span> <span class="math display">\[h_1=1,\qquad h_2=r,\qquad h_3=1 \]</span> <span class="math display">\[\nabla u=\frac{\partial u}{\partial r}\vec{e}_r+\frac{1}{r}\frac{\partial u}{\partial\theta}\vec{e}_{\theta}+\frac{\partial u}{\partial z}\vec{e}_z \]</span> <span class="math display">\[\nabla\cdot\vec{u}=\frac{1}{r}\frac{\partial}{\partial r}(ru_1)+\frac{1}{r}\frac{\partial u_2}{\partial\theta}+\frac{\partial u_3}{\partial z} \]</span> <span class="math display">\[\nabla\times\vec{u}=\frac{1}{r}\begin{vmatrix}
\vec{e}_r &r\vec{e}_{\theta} &\vec{e}_z\\
\frac{\partial}{\partial r} &\frac{\partial}{\partial\theta} &\frac{\partial}{\partial z}\\
u_1 &ru_2 &u_3
\end{vmatrix} \]</span> <span class="math display">\[\Delta u=\frac{1}{r}\frac{\partial}{\partial r}\Big(r\frac{\partial u}{\partial r}\Big)+\frac{1}{r^2}\frac{\partial^2 u}{\partial\theta^2}+\frac{\partial^2 u}{\partial z^2} \]</span></p>
]]></content>
<tags>
<tag>tensor</tag>
<tag>fluid mechanism</tag>
</tags>
</entry>
<entry>
<title>VSCode -- Remote Development using SSH</title>
<url>/2022/04/12/SSH-connect/</url>
<content><![CDATA[<p>用ssh远程连接服务器运行代码，虽然vim编辑能力很强大，加上各种配置和插件用起来也不错，但总归比不上vscode。 vscode也可以用vim的方式编辑，高亮、智能补全、调试等功能更加强大，是非常完美的编辑器了。 所以希望通过远程连接插件，利用服务器性能的同时，实现便捷的本地vscode编辑。</p>
<p>参考链接</p>
<blockquote>
<p>https://code.visualstudio.com/docs/remote/ssh#<br />
https://docs.microsoft.com/zh-cn/windows-server/administration/openssh/openssh_install_firstuse<br />
https://www.cnblogs.com/yyds/p/6992125.html</p>
</blockquote>
<span id="more"></span>
<h1 id="ssh-connect">SSH connect</h1>
<h2 id="本地和远程服务器安装ssh-powershell">本地和远程服务器安装ssh (PowerShell)</h2>
<p>管理员身份运行powershell <figure class="highlight powershell"><table><tr><td class="code"><pre><span class="line"><span class="built_in">Get-WindowsCapability</span> <span class="literal">-Online</span> | <span class="built_in">Where-Object</span> Name <span class="operator">-like</span> <span class="string">'OpenSSH*'</span></span><br></pre></td></tr></table></figure> 若均未安装 <figure class="highlight powershell"><table><tr><td class="code"><pre><span class="line">Name : OpenSSH.Client~~~~<span class="number">0.0</span>.<span class="number">1.0</span></span><br><span class="line">State : NotPresent</span><br><span class="line"></span><br><span class="line">Name : OpenSSH.Server~~~~<span class="number">0.0</span>.<span class="number">1.0</span></span><br><span class="line">State : NotPresent</span><br></pre></td></tr></table></figure> 根据需要安装 <figure class="highlight powershell"><table><tr><td class="code"><pre><span class="line"><span class="comment"># Install the OpenSSH Client</span></span><br><span class="line"><span class="built_in">Add-WindowsCapability</span> <span class="literal">-Online</span> <span class="literal">-Name</span> OpenSSH.Client~~~~<span class="number">0.0</span>.<span class="number">1.0</span></span><br><span class="line"></span><br><span class="line"><span class="comment"># Install the OpenSSH Server</span></span><br><span class="line"><span class="built_in">Add-WindowsCapability</span> <span class="literal">-Online</span> <span class="literal">-Name</span> OpenSSH.Server~~~~<span class="number">0.0</span>.<span class="number">1.0</span></span><br></pre></td></tr></table></figure> 两者都会返回以下输出 <figure class="highlight powershell"><table><tr><td class="code"><pre><span class="line">Path :</span><br><span class="line">Online : True</span><br><span class="line">RestartNeeded : False</span><br></pre></td></tr></table></figure></p>
<h2 id="ssh-密钥登录">ssh 密钥登录</h2>
<h3 id="在服务器生成公私密钥">在服务器生成公私密钥</h3>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ cd ~/.ssh</span><br><span class="line">$ ssh-keygen -t rsa -b 4096</span><br><span class="line">Generating public/private rsa key pair.</span><br><span class="line">Enter file in which to save the key (/home/[username]/.ssh/id_rsa): [keyname]</span><br><span class="line">Enter passphrase (empty for no passphrase): [passphrase or directly enter]</span><br><span class="line">Enter same passphrase again:</span><br><span class="line">Your identification has been saved in test</span><br><span class="line">Your public key has been saved in test.pub</span><br><span class="line">The key fingerprint is:</span><br><span class="line">SHA256:[xxxxxx]</span><br><span class="line">The key's randomart image is:</span><br><span class="line">+---[RSA 4096]----+</span><br><span class="line">| ....+o ..O|</span><br><span class="line">| . +o... +o|</span><br><span class="line">| . =.. .=|</span><br><span class="line">| . + o oo... .o|</span><br><span class="line">| o + *S * o |</span><br><span class="line">| E * * o + . |</span><br><span class="line">| . . O . . o.. |</span><br><span class="line">| o o . ....|</span><br><span class="line">| . oo*|</span><br><span class="line">+----[SHA256]-----+</span><br><span class="line"></span><br><span class="line">$ ls</span><br><span class="line">[keyname] [keyname].pub</span><br></pre></td></tr></table></figure>
<ul>
<li>注：
<ul>
<li>公私钥应在服务器生成，这样不同的本地机都能用</li>
<li>服务器应该同时有公钥和私钥，否则登录时可能报错</li>
</ul>
<code>load pubkey [path]: invalid format</code></li>
</ul>
<h3 id="在服务器安装公钥">在服务器安装公钥</h3>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ cd ~/.ssh</span><br><span class="line">$ cat [keyname].pub >> authorized_keys</span><br><span class="line">$ chmod 600 authorized_keys # 保证文件权限正确</span><br><span class="line">$ chmod 700 ~/.ssh</span><br></pre></td></tr></table></figure>
<p>编辑<code>/etc/ssh/sshd_config</code> <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">PubkeyAuthentication yes # 通常默认已经是此设置</span><br></pre></td></tr></table></figure></p>
<h3 id="下载私钥到本地连接ssh">下载私钥到本地，连接ssh</h3>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ssh -i [keyname] [hostname]@[ip] -p [port] # ssh密钥连接</span><br><span class="line"># 若是初次连接</span><br><span class="line">The authenticity of host '[[host]]:[port] ([[host]]:[port])' can't be established.</span><br><span class="line">ECDSA key fingerprint is SHA256:[xxxx].</span><br><span class="line">Are you sure you want to continue connecting (yes/no/[fingerprint])? yes</span><br><span class="line">Warning: Permanently added '[[host]]:[port]' (ECDSA) to the list of known hosts.</span><br><span class="line">[hostname]@[server name]:~$</span><br></pre></td></tr></table></figure>
<h1 id="vscode-设置">VSCode 设置</h1>
<ul>
<li>安装插件 <code>Remote - SSH</code></li>
<li><code>F1</code> => <code>Remote - SSH: Open SSH Configuration File...</code> => select file and edit <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">Host alias</span><br><span class="line"> HostName [ip]</span><br><span class="line"> User [username]</span><br><span class="line"> Port [port]</span><br><span class="line"> IdentityFile [path of keyname]</span><br></pre></td></tr></table></figure></li>
<li><code>F1</code> => <code>Remote - SSH: Connect to Host...</code> => select platform
<ul>
<li>注：若报错<code>ssh: Could not resolve hostname [hostname]: A non-recoverable error occurred during a database lookup.</code>， vscode插件的<code>config</code>文件有问题</li>
</ul></li>
<li>在远程ssh端安装需要的vscode插件</li>
</ul>
<h1 id="其他工具">其他工具</h1>
<p>推荐<code>MobaXterm</code>，连接稳定方便，支持各种终端，文件传输方便</p>
]]></content>
<tags>
<tag>code</tag>
<tag>linux</tag>
</tags>
</entry>
<entry>
<title>Latex Tips</title>
<url>/2022/04/13/latex-tips/</url>
<content><![CDATA[<p>整理一些好用的latex排版技巧 <span id="more"></span></p>
<h1 id="排版python代码">排版python代码</h1>
<p>参考链接</p>
<blockquote>
<p>https://www.zhihu.com/question/65508676/answer/232267619</p>
</blockquote>
<figure class="highlight tex"><table><tr><td class="code"><pre><span class="line"><span class="keyword">\usepackage</span>{listings, xcolor}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{lfonts}{</span><br><span class="line"> basicstyle = <span class="keyword">\footnotesize</span><span class="keyword">\ttfamily</span>,</span><br><span class="line"> stringstyle = <span class="keyword">\color</span>{purple},</span><br><span class="line"> keywordstyle = <span class="keyword">\color</span>{blue!60!black}<span class="keyword">\bfseries</span>,</span><br><span class="line"> commentstyle = <span class="keyword">\color</span>{olive}<span class="keyword">\scshape</span>,</span><br><span class="line">}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{lnumbers}{</span><br><span class="line"> numbers = left,</span><br><span class="line"> numberstyle = <span class="keyword">\tiny</span>,</span><br><span class="line"> numbersep = 1em,</span><br><span class="line"> firstnumber = 1,</span><br><span class="line"> stepnumber = 1,</span><br><span class="line">}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{llayout}{</span><br><span class="line"> breaklines = true,</span><br><span class="line"> tabsize = 2,</span><br><span class="line"> columns = flexible,</span><br><span class="line">}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{lgeometry}{</span><br><span class="line"> xleftmargin = 20pt,</span><br><span class="line"> xrightmargin = 0pt,</span><br><span class="line"> frame = tb,</span><br><span class="line"> framesep = <span class="keyword">\fboxsep</span>,</span><br><span class="line"> framexleftmargin = 20pt,</span><br><span class="line">}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{lgeneral}{</span><br><span class="line"> style = lfonts,</span><br><span class="line"> style = lnumbers,</span><br><span class="line"> style = llayout,</span><br><span class="line"> style = lgeometry,</span><br><span class="line">}</span><br><span class="line"><span class="keyword">\lstdefinestyle</span>{python}{</span><br><span class="line"> language = {Python},</span><br><span class="line"> style = lgeneral,</span><br><span class="line">}</span><br><span class="line"></span><br><span class="line"><span class="keyword">\begin</span>{document}</span><br><span class="line"><span class="keyword">\begin</span>{lstlisting}[style = python]</span><br><span class="line">for x in range(10):</span><br><span class="line"> pass</span><br><span class="line"><span class="keyword">\end</span>{lstlisting}</span><br><span class="line"><span class="keyword">\end</span>{document}</span><br></pre></td></tr></table></figure>
<h1 id="表格过宽的处理办法">表格过宽的处理办法</h1>
<figure class="highlight tex"><table><tr><td class="code"><pre><span class="line"><span class="keyword">\documentclass</span>{article}</span><br><span class="line"><span class="keyword">\usepackage</span>{graphicx}</span><br><span class="line"><span class="keyword">\begin</span>{document}</span><br><span class="line"><span class="keyword">\begin</span>{table}[!htb]</span><br><span class="line"> <span class="keyword">\centering</span></span><br><span class="line"> <span class="keyword">\caption</span>{Imagine that this is a table.}</span><br><span class="line"> <span class="keyword">\label</span>{tab:resized}</span><br><span class="line"> <span class="keyword">\resizebox</span>{<span class="keyword">\linewidth</span>}{!}{</span><br><span class="line"> <span class="comment">% table content</span></span><br><span class="line"> }</span><br><span class="line"><span class="keyword">\end</span>{table}</span><br></pre></td></tr></table></figure>
]]></content>
<tags>
<tag>code</tag>
<tag>latex</tag>
</tags>
</entry>
<entry>
<title>Linux Commands</title>
<url>/2022/04/13/linux-commands/</url>
<content><![CDATA[<p>整理了一些用到的比较好用的linux命令 <span id="more"></span></p>
<h1 id="后台运行">后台运行</h1>
<h2 id="后台运行保存运行日志">后台运行，保存运行日志</h2>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ nohup [your command] </dev/null >logfile 2>&1 &</span><br></pre></td></tr></table></figure>
<ul>
<li><code>nohup</code> 挂起命令<code>[your command]</code>在后台运行，不会被<code>ctrl C</code>终止</li>
<li><code><dev/null</code> 标准输入设为空</li>
<li><code>>logfile</code> 标准输出重定向到文件<code>logfile</code></li>
<li><code>2>&1</code> 标准错误输出重定向到标准输出，会一起保存在<code>logfile</code></li>
</ul>
<h2 id="查看日志">查看日志</h2>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ tail -f logfile</span><br></pre></td></tr></table></figure>
<h2 id="查看和终止后台进程">查看和终止后台进程</h2>
<figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ps -aux | grep [search content]</span><br><span class="line">$ kill -9 [process id]</span><br></pre></td></tr></table></figure>
<h1 id="查看文件夹内文件信息">查看文件夹内文件信息</h1>
<ul>
<li>当前目录下文件数量（不包括子目录中的文件） <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ls -l | grep "_" | wc -l</span><br></pre></td></tr></table></figure></li>
<li>当前目录下以<code>aabb</code>开头的文件数量（不包括子目录中的文件） <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ls -l aabb* | grep "_" | wc -l</span><br></pre></td></tr></table></figure></li>
<li>当前目录下文件数量（包括子目录中的文件） <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ls -lR | grep "_" | wc -l</span><br></pre></td></tr></table></figure></li>
<li>当前目录下文件夹数量（不包括子目录中的文件夹） <figure class="highlight plaintext"><table><tr><td class="code"><pre><span class="line">$ ls -l | grep "d" | wc -l</span><br></pre></td></tr></table></figure></li>
</ul>
]]></content>
<tags>
<tag>code</tag>
<tag>linux</tag>
</tags>
</entry>
</search>