on the Dirac equation, expands the ${\gamma}^{\mu}{\mathrm{\partial}}_{\mu}$ term as:

${\gamma}^{\mu}{\mathrm{\partial}}_{\mu}={\gamma}^{0}\frac{\mathrm{\partial}}{\mathrm{\partial}t}+\overrightarrow{\gamma}\cdot \overrightarrow{\mathrm{\nabla}}$

where $\overrightarrow{\gamma}=({\gamma}^{1},{\gamma}^{2},{\gamma}^{3})$, but to my knowledge,

${\gamma}^{\mu}{\mathrm{\partial}}_{\mu}={\gamma}^{\mu}{\eta}_{\mu \nu}{\mathrm{\partial}}^{\nu}={\gamma}^{0}\frac{\mathrm{\partial}}{\mathrm{\partial}t}-\overrightarrow{\gamma}\cdot \overrightarrow{\mathrm{\nabla}}$

using the convention ${\eta}_{\mu \nu}=\mathrm{diag}(+,-,-,-)$.