trjhtr

Using

iiint

seems to work fine to generator a triple integral. However, it doesn't seem to work if I include

includemathpazo

in the preamble. Any solutions for this? I am using the Palatino font for the main text. The MWE is as follows:

documentclass[a4paper,10pt,reqno,oldfontcommands]memoir
usepackage[T1]fontenc
usepackagefouriernc
usepackagemathpazo
usepackage[centertags]amsmath 
usepackageamssymb 
usepackageamsthm 
usepackagemathrsfs 

newcommand*difmathop!mathrmd 

begindocument

sectionA Section

This should be a triple integral:
beginalign
iiintlimits_V dif V , (nablaphi)^2 = 0.
labeleqn:triple_integral
endalign

enddocument

The mathpazo package, which provides Palatino-clone text and math fonts, hasn't been updated in a long time. As you've discovered, it also lacks macros to generate double- and triple-integral symbols.

An up-to-date alternative to mathpazo is the newpxtext and newpxmath pair of packages. A side-benefit of loading newpxmath is that it provides a iiint macro by default -- as well as several other, compatibly styled integral symbols.

Since newpxmath loads the amsmath package automatically (if it hasn't been loaded already), there's no need to load amsmath in separate instruction. Likewise, unless the noamssymbols option is set, the newpxmath package loads the pxfonts version of amssymb -- no need to load the amssymb package separately either

documentclassarticle
usepackagenewpxtext,newpxmath
begindocument
$int oint iint iiint displaystyleint oint iint iiint$

The quick brown fox jumps over the lazy dog.
enddocument

Addendum, posted after the OP provided a full MWE: The newpxmath and newpxtext packages must be loaded after amsmath.

documentclass[a4paper]memoir
usepackage[T1]fontenc
%usepackagefouriernc % not compatible with Palatino
usepackageamsmath % is loaded automatically by 'newpxmath' package
%usepackageamssymb % no need to load this package either
usepackageamsthm 
usepackagemathrsfs 
usepackagenewpxtext,newpxmath

newcommand*difmathop!mathrmd 

begindocument
setcounterchapter2 % just for this example
sectionA Section

This should be a triple integral:
beginequation labeleqn:triple_integral
iiintlimits_V ! dif V , (nablaphi)^2 = 0.
endequation
enddocument

Using xelatex or lualatex makes things easier:

documentclass[a4paper]memoir
usepackageamsthm 
usepackageunicode-math
setmainfontTeX Gyre Pagella
setmathfontTeX Gyre Pagella Math

newcommand*difmathop!mathrmd 

begindocument
 setcounterchapter2 % just for this example
 sectionA Section
 This should be a triple integral:
 beginequation labeleqn:triple_integral
 iiintlimits_V ! dif V , (nablaphi)^2 = 0.
 endequation
enddocument

The error is in loading fouriernc, which is for using New Century Schoolbook, rather than Palatino. Load either it or mathpazo, not both.

The centertags option is active by default for amsmath.

documentclass[a4paper,10pt,reqno,oldfontcommands]memoir
usepackage[T1]fontenc
usepackagemathpazo
usepackageamsmath
usepackageamssymb
usepackageamsthm
usepackagemathrsfs

newcommand*difmathop!mathrmd

begindocument

sectionA Section

This should be a triple integral:
beginequation
iiintlimits_V (nablaphi)^2 dif V = 0.
labeleqn:triple_integral
endequation

enddocument

A different option is to use newpx (amsthm should be loaded before it) and amssymb should be removed.

documentclass[a4paper,10pt,reqno,oldfontcommands]memoir
usepackage[T1]fontenc
usepackageamsthm
usepackagenewpxtext,newpxmath
usepackagemathrsfs

newcommand*difmathop!mathrmd

begindocument

sectionA Section

This should be a triple integral:
beginequation
iiintlimits_V (nablaphi)^2 dif V = 0.
labeleqn:triple_integral
endequation

enddocument

A good. modern replacement for mathpple/mathpazo/pxfonts/newpxmath is Asana Math, which was originally based on Young Ryu’s pxfonts.

documentclass[a4paper,10pt,reqno,oldfontcommands]memoir
usepackage[centertags]amsmath
usepackageunicode-math

defaultfontfeaturesScale=MatchLowercase
setmainfontTeX Gyre Pagella[Scale = 1.0, Ligatures=Common, TeX]
setmathfontAsana Math
% If you’re using mathrsfs, you probably want to use mathscr, but you also
% load amssymb. The following line sets up a mathcal and mathbfcal similar
% to the ones from ammsymb, in addition to mathscr and mathbfscr similar to
% rsfs.
setmathfont[range=cal,bfcal, Alternate]Asana Math

% URW Classico is a free clone of Optima, which matches the free clones of
% Palatino well.
%setsansfontURWClassico[
% UprightFont = *-Regular ,
% BoldFont = *-Bold ,
% ItalicFont = *-Italic ,
% BoldItalicFont = *-BoldItalic ,
% Extension = .ttf ]

%setmonofontInconsolata % A humanist monospaced font.

newcommand*difmathopmathrmd 

begindocument

sectionA Section

This should be a triple integral:
beginalign
iiintlimits_V dif V , (nablaphi)^2 = 0.
labeleqn:triple_integral
endalign

enddocument

For comparison, mathpazo with a double integral:

documentclass[a4paper,10pt,reqno,oldfontcommands]memoir
usepackage[T1]fontenc
usepackagetextcomp
usepackage[utf8]inputenc % The default since 2018.

usepackage[centertags]amsmath
usepackageamssymb
usepackagemathrsfs
usepackagemathpazo

newcommand*difmathopmathrmd 

begindocument

sectionA Section

This should be a triple integral:
beginalign
iintlimits_V dif V , (nablaphi)^2 = 0.
labeleqn:triple_integral
endalign

enddocument

My recommendation is to use unicode-math whenever you can, and legacy font packages when you have to. Any modern math font has more symbols with more consistency and more flexibility than any combination of legacy packages. If you’re stuck with PDFTeX, I suggest replacing usepackagemathpazo with usepackagenewpxtext, newpxmath.