Write more beautiful math equations, like a professional book

You don't mention how the ugly-looking "r with arrow above" was created, but it looks like it was done with \overset{r}{\to}. If one uses \vec instead, one gets more normal looking arrows.

I would also recommend using "dotless" i and j unit vectors.

\documentclass{article}
\usepackage{newpxmath}
\begin{document}
$ d\vec{r} = dx\hat{\imath} + dy\hat{\jmath} + dz\hat{k} $
\end{document}

A first note: The equations shown below represent an identical copy of the second expression shown by the OP. Since they are not correct, according to the accepted vector notation, I apologize in advance.

From the first equation it can be easily recognized that the \usepackage{palatino} and \usepackage{mathpazo}has been used, because of the typical italic setting, so with a basic MWE the bold faced variant could be defined as you like:

\documentclass{article}
%
\usepackage{amsmath}
\usepackage{palatino}
\usepackage{mathpazo}
%
\begin{document}
\[
\mathrm{d}\mathbf{r}=%
\mathrm{d}x\hat{\mathbf{i}}+%
\mathrm{d}y\hat{\mathbf{j}}+%
\mathrm{d}z\hat{\mathbf{k}}
\]
\end{document}

Here is the (I hope it's beautiful enough) example output:

---

EDIT:

This workaround has been done by substituting mathpazo with newtxmath:

\documentclass{article}
%
\usepackage{amsmath}
\usepackage{palatino}
\usepackage{newtxmath}
%
\begin{document}
\[
\mathrm{d}\mathbf{r}=%
\mathrm{d}x\hat{\mathbf{i}}+%
\mathrm{d}y\hat{\mathbf{j}}+%
\mathrm{d}z\hat{\mathbf{k}}
\]
\end{document}

As you can see the plus and equal sign are smaller by default. To change directly these operators inside the old font would require some kind of sorcery that at the end won't match a default setting.

\documentclass[preview,border=12pt]{standalone}
\usepackage{amsmath}
\usepackage[d]{esvect}

\newcommand\dd{\textrm{d}}

\newcommand\xhat{\,\hat{x}}
\newcommand\yhat{\,\hat{y}}
\newcommand\zhat{\,\hat{z}}

\newcommand\ihat{\,\hat{\imath}}
\newcommand\jhat{\,\hat{\jmath}}
\newcommand\khat{\,\hat{k}}

\newcommand\xu{\,\boldsymbol{x}}
\newcommand\yu{\,\boldsymbol{y}}
\newcommand\zu{\,\boldsymbol{z}}

\newcommand\iu{\,\boldsymbol{i}}
\newcommand\ju{\,\boldsymbol{j}}
\newcommand\ku{\,\boldsymbol{k}}

\begin{document}
\abovedisplayskip=0pt\relax
\begin{align}
\dd\vv{r} &=\dd x\xhat+\dd y\yhat+\dd z\zhat\\
\dd\vv{r} &=\dd x\ihat+\dd y\jhat+\dd z\khat\\
\dd\boldsymbol{r} &=\dd x\xu+\dd y\yu+\dd z\zu\\
\dd\boldsymbol{r} &=\dd x\iu+\dd y\ju+\dd z\ku
\end{align}
\end{document}