Boxing Multiple Lines in Split

Question

I want to put a box around the final two lines in this

\begin{equation}
\begin{split}
\left\langle\vec{u}\right\rangle&=\frac{1}{4}\Re\left(\epsilon_0\vec{\overset{\sim}{E}}\cdot\vec{\overset{\sim}{E}}^*+\frac{1}{\mu_0}\vec{\overset{\sim}{B}}\cdot\vec{\overset{\sim}{B}}^*\right) \\
&=\frac{1}{4}\left[\epsilon_0\left(\frac{\pi\omega B_0}{\left(\frac{\omega}{c}\right)^2-k^2}\right)^2\left[\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)+\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]\right. \\
&\qquad+\frac{1}{\mu_0}\left(\frac{\pi kB_0}{\left(\frac{\omega}{c}\right)^2-k^2}\right)^2\left[\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)+\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)\right. \\
&\qquad\qquad\left.\left(\frac{\left(\frac{\omega}{c}\right)^2-k^2}{\pi k}\right)\cos^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right] \\

%Box starting here

&=\left(\frac{\pi B_0}{2\left[\left(\frac{\omega}{c}\right)^2-k^2\right]}\right)^2\left[\left(\epsilon_0\omega+\frac{k}{\mu_0}\right)\left[\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)\right.\right. \\
&\qquad\left.\left.+\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]+\left(\frac{\left(\frac{\omega}{c}\right)^2-k^2}{\pi\mu_0k}\right)\cos^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]

%and ending here

\end{split}
\end{equation}

I want it to pretty much look like this

\begin{equation}
\begin{split}
\left\langle\vec{S}\right\rangle&=\frac{1}{2\mu_0}\Re\left(\vec{\overset{\sim}{E}}\times\vec{\overset{\sim}{B}}^*\right) \\
&=\frac{1}{2\mu_0}\Re\left(\frac{i\pi B_0}{\left(\frac{\omega}{c}\right)^2-k^2}\left[\left\langle\frac{-n\omega}{b}\left[\cos\left(\frac{m\pi}{a}x\right)\sin\left(\frac{n\pi}{b}y\right)\right],\frac{m\omega}{a}\left[\sin\left(\frac{m\pi}{a}x\right)\cos\left(\frac{n\pi}{b}y\right)\right],0\right\rangle\right.\right. \\
&\qquad\times\left\langle\frac{mk}{a}\left[\sin\left(\frac{m\pi}{a}x\right)\cos\left(\frac{n\pi}{b}y\right)\right],\frac{nk}{b}\left[\cos\left(\frac{m\pi}{a}x\right)\sin\left(\frac{n\pi}{b}y\right)\right],\right. \\
&\qquad\qquad\left.\left.\left.\frac{-i\left[\left(\frac{\omega}{c}\right)^2-k^2\right]}{\pi}\cos\left(\frac{m\pi}{a}x\right)\cos\left(\frac{n\pi}{b}y\right)\right\rangle\right]\right) \\
&=\frac{1}{2\mu_0}\Re\left(\frac{i\pi B_0}{\left(\frac{\omega}{c}\right)^2-k^2}\left\langle\frac{-im\omega\left[\left(\frac{\omega}{c}\right)^2-k^2\right]}{\pi a}\sin\left(\frac{m\pi}{a}x\right)\cos\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right),\right.\right. \\
&\qquad\frac{in\omega\left[\left(\frac{\omega}{c}\right)^2-k^2\right]}{\pi b}\cos^2\left(\frac{m\pi}{a}x\right)\sin\left(\frac{n\pi}{b}y\right)\cos\left(\frac{n\pi}{b}y\right), \frac{-n^2\omega k}{b^2}\left[\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)\right] \\
&\qquad\qquad\left.\left.-\frac{m^2\omega k}{a^2}\left[\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]\right\rangle\right) \\
&\fbox{$=\frac{\omega B_0}{2\mu_0}\left\langle\frac{m}{a}\sin\left(\frac{m\pi}{a}x\right)\cos\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right),\frac{n}{b}\cos^2\left(\frac{m\pi}{a}x\right)\sin\left(\frac{n\pi}{b}y\right)\cos\left(\frac{n\pi}{b}y\right),0\right\rangle$}
\end{split}
\end{equation}

I've tried using \fbox, but I cannot seem to get it to span multiple lines keeping my paragraphing and tabs. Also sorry for the messy code!

Answer 1

The amsmath package provides a macro called \boxed for just this purposes. Note that I encased an aligned environment inside the \boxed directive. And I also replaced all instances of \left(\frac{\omega}{c}\right) with (\omega/c) to achieve a more compact appearance of the formulas.

In the following solution, the = symbol is not inside the box. If it has to be inside the box, you'll need to replace

%Box starting here
&=\boxed{\begin{aligned}[t]
&\biggl(\frac{\pi B_0}{2\left[(\omega/c)^2-k^2\right]}\biggr)^{\!2}

with

%Box starting here
&\boxed{\begin{aligned}
&=\biggl(\frac{\pi B_0}{2\left[(\omega/c)^2-k^2\right]}\biggr)^{\!2}

in the code given below.

\documentclass{article}
\usepackage[letterpaper,margin=1in]{geometry} % set page parameters appropriately
\usepackage{amsmath}
\usepackage{mleftright} % for tightly-spaced \left...\right constructs
\mleftright

\begin{document}
\begin{equation}
\begin{split}
\langle\vec{u}\mkern1.5mu\rangle
&=\frac{1}{4}\Re\Bigl( \epsilon_0\vec{\widetilde{E}}\cdot\vec{\widetilde{E}}^*
 +\frac{1}{\mu_0}\vec{\widetilde{B}}\cdot\vec{\widetilde{B}}^* \Bigr) \\
&=\frac{1}{4}
 \Biggl[
 \epsilon_0
 \left(\frac{\pi\omega B_0}{(\omega/c)^2-k^2}\right)^{\!2}
 \left[\left(\frac{n}{b}\right)^2\cos^2
 \left(\frac{m\pi}{a}x\right)\sin^2
 \left(\frac{n\pi}{b}y\right)
+\left(\frac{m}{a}\right)^2\sin^2
 \left(\frac{m\pi}{a}x\right)\cos^2
 \left(\frac{n\pi}{b}y\right)\right] \\
&\qquad+\frac{1}{\mu_0}
 \left(\frac{\pi kB_0}{(\omega/c)^2-k^2}\right)^{\!2}
 \left[\left(\frac{m}{a}\right)^2\sin^2
 \left(\frac{m\pi}{a}x\right)\cos^2
 \left(\frac{n\pi}{b}y\right)
+\left(\frac{n}{b}\right)^2\cos^2
 \left(\frac{m\pi}{a}x\right)\sin^2
 \left(\frac{n\pi}{b}y\right)\right. \\
&\qquad\qquad
 \left(\frac{(\omega/c)^2-k^2}{\pi k}\right)\cos^2 
 \left(\frac{m\pi}{a}x\right)\cos^2
 \left(\frac{n\pi}{b}y\right) 
 \Biggr] \\
%Box starting here
&=\boxed{\begin{aligned}[t]
&\biggl(\frac{\pi B_0}{2\left[(\omega/c)^2-k^2\right]}\biggr)^{\!2}
 \Biggl[
 \left(\epsilon_0\omega+\frac{k}{\mu_0}\right)
 \biggl[\left(\frac{n}{b}\right)^2\cos^2
 \left(\frac{m\pi}{a}x\right)\sin^2
 \left(\frac{n\pi}{b}y\right) \\
&\qquad+\left(\frac{m}{a}\right)^2\sin^2
 \left(\frac{m\pi}{a}x\right)\cos^2
 \left(\frac{n\pi}{b}y\right)\biggr]
+\left(\frac{(\omega/c)^2-k^2}{\pi\mu_0k}\right)\cos^2
 \left(\frac{m\pi}{a}x\right)\cos^2
 \left(\frac{n\pi}{b}y\right)
 \Biggr] 
\end{aligned}}
%and ending here
\end{split}
\end{equation}
\end{document}

Answer 2

The \Aboxed command from mathtools alows for including the alignment point in a framebox. To enclose several lines I use the multlined environment from the same package, and aligned for the lines above:

\documentclass{article}
\usepackage{geometry} 
\usepackage{mathtools}

\begin{document}

\begin{equation}
\begin{split}
\left\langle\vec{u}\right\rangle&=\frac{1}{4}\Re\left(\epsilon_0\vec{\overset{\sim}{E}}\cdot\vec{\overset{\sim}{E}}^*+\frac{1}{\mu_0}\vec{\overset{\sim}{B}}\cdot\vec{\overset{\sim}{B}}^*\right) \\
&=\begin{alignedat}[t]{2} \frac{1}{4}\Biggl[& \epsilon_0\left(\frac{\pi\omega B_0}{\left(\frac{\omega}{c}\right)^2-k^2}\right)^{\mkern-5mu 2} \left[\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)+\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right] & & \\
 & +\frac{1}{\mu_0}\left(\frac{\pi kB_0}{\left(\frac{\omega}{c}\right)^2 - k^2}\right)^{\mkern-5mu 2} \biggl[\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\\%
 & & \mathllap{+\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)
\left(\frac{\left(\frac{\omega}{c}\right)^2-k^2}{\pi k}\right)\cos^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\Biggr]} &
\end{alignedat}\\
%Box starting here
\Aboxed{&=\begin{multlined}[t] \left(\frac{\pi B_0}{2\left[\left(\frac{\omega}{c}\right)^2-k^2\right]}\right)^{\mkern-5mu 2} \left[\left(\epsilon_0\omega+\frac{k}{\mu_0}\right)\left[\left(\frac{n}{b}\right)^2\cos^2\left(\frac{m\pi}{a}x\right)\sin^2\left(\frac{n\pi}{b}y\right)\right.\right. \\
\left.\left.+\left(\frac{m}{a}\right)^2\sin^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]+\left(\frac{\left(\frac{\omega}{c}\right)^2-k^2}{\pi\mu_0k}\right)\cos^2\left(\frac{m\pi}{a}x\right)\cos^2\left(\frac{n\pi}{b}y\right)\right]
\end{multlined}}
%and ending here
\end{split}
\end{equation}

 \end{document}