The previous material derivative equation is a vector equation. As with all vector equations, it may be written in scalar component equations. Relative to an \( xyz \) coordinate system, the scalar components of the equation are written:

\[ a_{x_p} = \frac{D v_x}{D t} = v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z} + \frac{\partial v_x}{\partial t} \] \[ a_{y_p} = \frac{D v_y}{D t} = v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z} + \frac{\partial v_y}{\partial t} \] \[ a_{z_p} = \frac{D v_z}{D t} = v_x \frac{\partial v_z}{\partial x} + v_y \frac{\partial v_z}{\partial y} + v_z \frac{\partial v_z}{\partial z} + \frac{\partial v_z}{\partial t} \]