\DeclareMathOperator is designed to create commands that should typeset operator names such as sin and lim. Some of these are already defined in base TeX or LaTeX so one writes 2\sin\theta
instead of 2sin\theta
giving correct spacing and font. If you need an operator of this type that is not predefined, then you create it with \DeclareMathOperator, e.g. the space of endomorphisms of a vector space is written \End V
but you need to make the definition \DeclareMathOperator{\End}{End} first: a minimal working example is
\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\End}{End}
\begin{document}
\( \End V \)
\end{document}
\newcommand is much more general and can be used to define direct short cuts or more complicated macros. So for example if you find youself writing \left.\frac d{dt}\right|_{t=0} many times in your document you can package this up as a single command \dtzero via \newcommand{\dtzero}{\left.\frac d{dt}\right|_{t=0}} and just type \dtzero each time instead.
I would usually recommend reserving \DeclareMathOperator for the use described above and using \newcommand in most other situations. To get the effect of \DeclareMathOperator in a one-off situation, or inside a \newcommand, you use \operatorname; so you can write \operatorname{End}V for the above example.
Finally, one should note that there is a starred version \DeclareMathOperator*. This is used for defining operators that have limits typeset beneath them instead of to the right (at least when in a display). For example
Similarly there is the starred variant \operatorname*.
\documentclass{article}
\usepackage{amsmath}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator*{\Max}{Max}
\begin{document}
\begin{displaymath}
\Max_{x\in A} f(x) \qquad \End_R V
\end{displaymath}
\end{document}
Remark. The above code samples load the amsmath package. Strictly speaking all you need is the amsopn package, which amsmath reads in automatically. Alternatively, one can load mathtools which is a modern package building further on amsmath.
The \mathbf around \Div does nothing, because \Div is translated internally to something like "choose the \mathrm font and typeset ‘Div’”. Math fonts don't inherit features from the context: \mathrm always corresponds to the upright text font in medium weight (or boldface) if \boldmath is in force.
You can check that
$\mathbf{\mathrm{x}}$
produces a medium weight ‘x’.
So you have to define
\DeclareMathOperator{\Div}{\mathbf{Div}}
\newcommand{\Divxk}{\Div_{X/k}}
The definition of \Div becomes “choose the \mathrm font and typeset ‘\mathbf{Div}’”, so, for the same reason as before, the boldface font will be chosen.
Best Answer
Quoting from this question,
DeclareMathOperatoris designed to create commands that should typeset operator names such assinandlim. The default font for these quantities is upright. If you want part of the operator name to be slanted (in the way variables are usually typeset), you have to specify this explicitly:On the other hand, I would guess that what you actually want is that only the 'z' itself is the operator name:
The part in parenthesis should be the argument to this operator. If you frequently use the same argument, you could define a short hand for this, using
\newcommand:Concerning your second question, if you encapsulate your definition with
\ensuremath, you can use your command both within or outside of math environments:With the last definition, both
$\zk$and\zk{}should give the same result.