You surely can define your shortcuts; indeed you should.
Let's make some examples. Suppose your document is full of Fourier transforms, for which you need a fancy F. Instead of writing every time
$\mathcal{F}(f)$
it's surely better to define a new command, say
\newcommand{\FT}{\mathcal{F}}
(choose any name you like), so that you can type
$\FT(f)$
and get the same result as before, with a big bonus! If you change your mind about the notation, you can simply modify the definition.
Another example. The "gradient" operator is not predefined; so you might want to have a command for it:
\DeclareMathOperator{\grad}{grad}
A different one; my preferred notation for vectors is, say, \mathbf{v}. However, since conventions are different, I never type vectors in that way, but prefer to have
\newcommand{\vect}[1]{\mathbf{#1}}
for the same reason as before; I might change my mind and want to modify the appearance, say for using bold italic; this would be accomplished just by saying
\usepackage{bm}
and changing the above into
\newcommand{\vect}[1]{\bm{#1}}
How do you organize this? Here's an example:
\documentclass[a4paper]{book}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % input encoding UTF-8
% Useful packages
\usepackage{amsmath,amssymb,amsthm}
% add all the packages you need
% Personal definitions
\newcommand{\FT}{\mathcal{F}} % Fourier transform operator
\DeclareMathOperator{\grad}{grad} % gradient
\newcommand{\vect}[1]{\mathbf{#1}} % vectors and matrices
\begin{document}
...
\end{document}
Add definitions while you find that they are useful for distinguishing logical units of your document.
Getting around the bug is easy: Add \setmathfont{lmodern-math.otf} after loading unicode-math. Btw: unicode-math can also be used with xelatex.
And if you want to declare definitions for utf8-input you can do it like this:
\documentclass{article}
\usepackage[utf8]{inputenc}
\DeclareUnicodeCharacter{222B}{\int}%U+222B= integral
\begin{document}
$∫f(x)$
\end{document}
Best Answer
You can use the
yhcmex10font that provides an extensible arc accent: