Using alphabets to label equations

Question

I have the following codes that label equations the usual way:

\begin{align}
  \partial^2 EU/\partial \alpha \partial \widebar{d}&= U'(\epsilon)[G_{\alpha}(\alpha,\theta,\epsilon) - \widebar{P} C_{\alpha}(\alpha,\theta)] - U'(0)[G_{\alpha}(\alpha,\theta,0) - C_{\alpha}(\alpha,\theta)], \label{eqn:21}
\end{align}
\begin{align}  
  \partial^2 EU/\partial \theta \partial \widebar{d} &= U'(\epsilon)[G_{\theta}(\alpha,\theta,\epsilon) - \widebar{P} C_{\theta}(\alpha,\theta)] + U'(0) [C_{\theta}(\alpha,\theta) - G_{\theta}(\alpha,\theta,0)]. \label{eqn:22}
\end{align}\\ 

Output:

Instead of numbers I want to label the equations as A.21 and A.22. Any suggestions on how I do that?

Here is what I have in my preamble:

\documentclass[12pt,reqno]{amsart}

% Packages
\usepackage[onehalfspacing]{setspace}
\usepackage{graphicx}
\usepackage{amssymb,amsthm}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{float}
\makeatletter
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
\newcommand*\widebar[1]{%
  \begingroup
  \def\mathaccent##1##2{%
    \rel@kern{0.8}%
    \overline{\rel@kern{-0.8}\macc@nucleus\rel@kern{0.2}}%
    \rel@kern{-0.2}%
  }%
  \macc@depth\@ne
  \let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
  \mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
  \macc@set@skewchar\relax
  \let\mathaccentV\macc@nested@a
  \macc@nested@a\relax111{#1}%
  \endgroup
}
\makeatother
\usepackage{cite}
%\usepackage{natbib}
%\bibpunct{(}{)}{;}{;}{,}{,}
\usepackage{xr-hyper}
\usepackage[
  colorlinks=true,
  citecolor=blue,
  urlcolor=blue,
  linkcolor=blue
]{hyperref}
\usepackage{bm}
\usepackage{fullpage}
\usepackage{ amssymb }
\usepackage{caption} 
\usepackage{forest}
\captionsetup{labelfont=normalfont,
              labelsep=colon}
\usepackage{mathabx}
\usepackage{subfig}
\usetikzlibrary{arrows.meta,shapes,positioning}
%
\tikzset{
    full/.style={circle,draw,inner sep=0, minimum size=1mm,fill=black},
    every node/.style={minimum height=5mm,font=\footnotesize}
}
%
\makeatletter
\def\subsection{\@startsection{subsection}{1}%
  \z@{.5\linespacing\@plus.7\linespacing}{-.5em}%
  {\normalfont\itshape}}
\def\@sect#1#2#3#4#5#6[#7]#8{%
  \edef\@toclevel{\ifnum#2=\@m 0\else\number#2\fi}%
  \ifnum #2>\c@secnumdepth \let\@secnumber\@empty
  \else \@xp\let\@xp\@secnumber\csname the#1\endcsname\fi
  \@tempskipa #5\relax
  \ifnum #2>\c@secnumdepth
    \let\@svsec\@empty
  \else
    \refstepcounter{#1}%
    \edef\@secnumpunct{%
      \ifdim\@tempskipa>\z@ % not a run-in section heading
        \@ifnotempty{#8}{.\@nx\enspace}%
      \else
        \@ifempty{#8}{.}{.\@nx\enspace}%
      \fi
    }%
    \@ifempty{#8}{%
      \ifnum #2=\tw@ \def\@secnumfont{\bfseries}\fi}{}%
    \protected@edef\@svsec{%
      \ifnum#2<\@m
        \@ifundefined{#1name}{}{%
          \ignorespaces\csname #1name\endcsname\space
        }%
      \fi
      \@seccntformat{#1}%
    }%
  \fi
  \ifdim \@tempskipa>\z@ % then this is not a run-in section heading
    \begingroup #6\relax
    \@hangfrom{\hskip #3\relax\@svsec}{\interlinepenalty\@M #8\par}%
    \endgroup
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  \else
  \def\@svsechd{#6\hskip #3\@svsec
    \@ifnotempty{#8}{\ignorespaces#8\unskip
       %\@addpunct.
       }%
    \ifnum#2>\@m \else \@tocwrite{#1}{#8}\fi
  }%
  \fi
  \global\@nobreaktrue
  \@xsect{#5}}
\makeatother

\pagestyle{plain}

\setlength{\parskip}{\baselineskip}
\setlength{\parindent}{12pt}

\setcounter{secnumdepth}{2}

\allowdisplaybreaks[4]
% Commenting/debugging
\let\IG\iffalse
\let\ENDIG\fi

%% Shortcuts
\newcommand{\td}[2]{\dfrac{d #1}{d #2}}
\newcommand{\std}[2]{\dfrac{d^2 #1}{d {#2}^2}}
\newcommand{\ctd}[3]{\dfrac{d^2 #1}{d #2 d #3}}

\newcommand{\pd}[2]{\dfrac{\partial #1}{\partial #2}}
\newcommand{\spd}[2]{\dfrac{\partial^2 #1}{\partial {#2}^2}}
\newcommand{\cpd}[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}}

\newcommand{\pdi}[2]{\partial #1/\partial #2}

\newcommand{\LR}{\Leftrightarrow}
\newcommand{\Lg}{\mathcal{L}}
\newcommand{\half}{\tfrac{1}{2}}
\newcommand{\eqp}{\phantom{=}}
\newcommand{\eqs}{\buildrel s \over =}
\newcommand{\me}{\mathrm{e}}
%begin codes for footnotes (symbols)
\makeatletter
\newcommand*{\myfnsymbolsingle}[1]{%
  \ensuremath{%
    \ifcase#1% 0
    \or % 1
      \dagger%   
    \or % 2
      1 
    \or % 3  
      2
    \or % 4   
      3
    \or % 5
      4
    \else % >= 6
      \@ctrerr  
    \fi
  }%   
}   
\makeatother

\newcommand*{\myfnsymbol}[1]{%
  \myfnsymbolsingle{\value{#1}}%
}

% remove upper boundary by multiplying the symbols if needed
\usepackage{alphalph}
\newalphalph{\myfnsymbolmult}[mult]{\myfnsymbolsingle}{}

\renewcommand*{\thefootnote}{%
  \myfnsymbolmult{\value{footnote}}%
}
%end codes for footnotes (symbols) 

Answer 1

Since the OP decided to have starred sections, the easiest way is to use

\renewcommand{\theequation}{A.\arabic{equation}}

but not recommended, since this harder to change.

If the equation number should be preceded by the section number or appendix number, then \renewcommand{\theequation}{\thesection.\arabic{equation}} is one way to do this.

Other alternative:

Just say \numberwithin{equation}{section}, this will do effectively the same

Note: The equations are too large or you have to enlarge the textwidth.

\documentclass{article}

%    \usepackage{mathtools} 
\usepackage{amsmath}
\usepackage{mathabx}
\begin{document}

\appendix
\section{First appendix}

\renewcommand{\theequation}{\thesection.\arabic{equation}}

\setcounter{equation}{21}

\begin{align}
  \partial^2 EU/\partial \alpha \partial \widebar{d}&= U'(\epsilon)[G_{\alpha}(\alpha,\theta,\epsilon) - \widebar{P} C_{\alpha}(\alpha,\theta)] - U'(0)[G_{\alpha}(\alpha,\theta,0) - C_{\alpha}(\alpha,\theta)], \label{eqn:21}
\end{align}
\begin{align}  
  \partial^2 EU/\partial \theta \partial \widebar{d} &= U'(\epsilon)[G_{\theta}(\alpha,\theta,\epsilon) - \widebar{P} C_{\theta}(\alpha,\theta)] + U'(0) [C_{\theta}(\alpha,\theta) - G_{\theta}(\alpha,\theta,0)]. \label{eqn:22}
\end{align} 

\end{document}

Answer 2

Another solution that uses the \newtagform and \usetagform, from mathtools, and the apptools package to automate thu use of these commands.

Additionally, I changed you partial derivatives to fractions, with the easy syntax of the esdiff package. I also loaded the geometry package to keep equations on one line, and grouped the two align environment in a single align.

\documentclass{article}

\usepackage{mathtools}
\usepackage{geometry}
\usepackage[thinc]{esdiff}
\makeatletter
\newcommand*\rel@kern[1]{\kern#1\dimexpr\macc@kerna}
\newcommand*\widebar[1]{%
\begingroup
\def\mathaccent##1##2{%
\rel@kern{0.8}%
\overline{\rel@kern{-0.8}\macc@nucleus\rel@kern{0.2}}%
\rel@kern{-0.2}%
}%
\macc@depth\@ne
\let\math@bgroup\@empty \let\math@egroup\macc@set@skewchar
\mathsurround\z@ \frozen@everymath{\mathgroup\macc@group\relax}%
\macc@set@skewchar\relax
\let\mathaccentV\macc@nested@a
\macc@nested@a\relax111{#1}%
\endgroup
}
\makeatother

\newtagform{appendix}{(A.}{)}
\usepackage{apptools}
\AtAppendix{\setcounter{equation}{0}\usetagform{appendix}}

\begin{document}
\setcounter{equation}{20}
\begin{align}
  \diffp{EU}{{\alpha} {\widebar{d}}} & = U'(\epsilon)\bigl(G_{\alpha}(\alpha,\theta,\epsilon) - \widebar{P} C_{\alpha}(\alpha,\theta)\bigr) - U'(0)\bigl(G_{\alpha}(\alpha,\theta,0) - C_{\alpha}(\alpha,\theta)\bigr), \label{eqn:21} \\
  \diffp{EU}{{\theta} {\widebar{d}}} & = U'(\epsilon)\bigl(G_{\theta}(\alpha,\theta,\epsilon) - \widebar{P} C_{\theta}(\alpha,\theta)\bigr) + U'(0) \bigl(C_{\theta}(\alpha,\theta) - G_{\theta}(\alpha,\theta,0)\bigr). \label{eqn:22}
\end{align}

\appendix
\setcounter{equation}{20}
\section
{A first appendix}Blah blah blah
\begin{align}
  \diffp{EU}{{\alpha} {\widebar{d}}} & = U'(\epsilon)\bigl(G_{\alpha}(\alpha,\theta,\epsilon) - \widebar{P} C_{\alpha}(\alpha,\theta)\bigr) - U'(0)\bigl(G_{\alpha}(\alpha,\theta,0) - C_{\alpha}(\alpha,\theta)\bigr), \label{eqn:21} \\
  \diffp{EU}{{\theta} {\widebar{d}}} & = U'(\epsilon)\bigl(G_{\theta}(\alpha,\theta,\epsilon) - \widebar{P} C_{\theta}(\alpha,\theta)\bigr) + U'(0) \bigl(C_{\theta}(\alpha,\theta) - G_{\theta}(\alpha,\theta,0)\bigr). \label{eqn:22}
\end{align}

\end{document}