amsmath's multline causes "Missing $ inserted" error

Question

I use the multline feature from amsmath to display an equation that spreads acroos multiple rows. I tried to do that like this:

\begin{multline}
    \label{eq:volume_boundaries}
    \int\limits_S \Gamma\nabla\phi\cdot\mathbf{n}\d{S} =
    \int\limits_{S_\text{n}} (\Gamma\nabla\phi\cdot\mathbf{n})_\text{n}\d{S_\text{n}} 
    +\int\limits_{S_\text{e}} (\Gamma\nabla\phi\cdot\mathbf{n})_\text{e}\d{S_\text{e}}\\
    +\int\limits_{S_\text{s}} (\Gamma\nabla\phi\cdot\mathbf{n})_\text{s}\d{S_\text{s}}
    +\int\limits_{S_\text{w}} (\Gamma\nabla\phi\cdot\mathbf{n})_\text{w}\d{S_\text{w}}
\end{multline}

However this leads to the error message Missing $ inserted. \end{multline}. What did I do wrong here?

Answer 1

The root cause of the problem you've encountered is that you're trying to use a text-mode-only command -- \d -- in math mode.

Rather than undefine and then redefine \d suitably, I think it's better to create a macro with a new name -- say, \diff -- to denote the "differential operator". E.g.,

\newcommand{\diff}{\mathop{}\!\mathrm{d}}

Observe that this command does not take an argument. Thus, I would encourage you to just write \diff S, not \diff{S}, as the latter might create the (misleading and inappropriate) impression that \diff is a macro that takes an argument.

I would also replace with all instances of \int\limits with just \int, as I can't see a (typographic) justification for elongating the equation in the vertical direction.

Finally, I think the equation is easier to read if you used an \equation/aligned combination instead of a multline environment.

\documentclass{article} % or some other suitable document class
\usepackage{amsmath}
\newcommand{\diff}{\mathop{}\!\mathrm{d}} % don't re-purpose the existing '\d' macro
\newcommand{\gnpn}{\Gamma\nabla\mkern-2mu \phi\cdot\mathbf{n}} % handy shortcut macro
\begin{document}

\begin{equation}\label{eq:volume_boundaries}
\begin{aligned}[b]
\int_S \gnpn \diff S 
    &=      \int_{S_\mathrm{n}}\!  (\gnpn)_{\mathrm{n}}\diff S_{\mathrm{n}} 
           +\int_{S_\mathrm{e}}\!  (\gnpn)_{\mathrm{e}}\diff S_{\mathrm{e}}\\
    &\quad +\int_{S_\mathrm{s}}\!  (\gnpn)_{\mathrm{s}}\diff S_{\mathrm{s}}
           +\int_{S_\mathrm{w}}\!\!(\gnpn)_{\mathrm{w}}\diff S_{\mathrm{w}}
\end{aligned}
\end{equation}
\end{document}