You have to use mathbf
to produce the upright math fonts.
Since they usually represent vectors, you can renew the \vec
command.
Also I created a derivative operator \drv
which prints an upright d
(as a derivative sign) keeping the spacing (at least to its right) that is present in the text
Here an implementation
\documentclass{article}
\usepackage{amsmath}
\renewcommand{\vec}[1]{\mathbf{#1}}
\DeclareRobustCommand*{\drv}{\mathop{}\!\mathrm{d}}
\begin{document}
\begin{equation}
\frac{\drv\vec{P}_k^-}{\drv t}=\vec{F}_x(\vec{m}_k^-(t),t,\vec{\theta})\vec{P}_k^- +\vec{P}_k^- \vec{F}_x^T(\vec{m}_k^-(t),t,\vec{\theta}) + \Sigma(\vec{m}_k^-(t),t,\vec{\theta})
\label{eq:Euler}
\end{equation}
\end{document}

Thanks to @GustavoMezzetti and @lblb for their insightful remarks
EDIT
Since the user asked, I provide a solution, which I don't know if optimal, concerning the bold math symbols.
The "normal" way in fact should be
Load bm
and upgreek
Substitute the greek letter with its upshape, ie theta
becomes uptheta
Call \bm{\uptheta}
instead of vec
If one wants to keep the vec
version, instead, has only to substitute theta with uptheta to obtain what he wants:
\documentclass{article}
\usepackage{amsmath,bm,mathtools,upgreek}
\renewcommand{\vec}[1]{\bm{\mathrm{#1}}}
\DeclareRobustCommand*{\drv}{\mathop{}\!\mathrm{d}}
\begin{document}
\begin{equation}
\frac{\drv\vec{P}_k^-}{\drv t}=\vec{F}_x(\vec{m}_k^-(t),t,\vec{\uptheta})\vec{P}_k^- +\vec{P}_k^- \vec{F}_x^T(\vec{m}_k^-(t),t,\vec{\uptheta}) + \Sigma(\vec{m}_k^-(t),t,\vec{\uptheta})
\label{eq:Euler}
\end{equation}
\end{document}
