$ 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} $
The previous result is often written in shorthand notation as $ \vec{a} = \frac{D \vec{V}}{D t} $
where the operator $ \frac{D( \ )}{D t} = \frac{\partial ( \ )}{\partial t} + v_x \frac{\partial ( \ )}{\partial x} + v_y \frac{\partial ( \ )}{\partial y} + v_z \frac{\partial ( \ )}{\partial z} $
It is termed the material derivative or substantial derivative. An often-used shorthand notation for the material derivative operator is: $ \frac{D( \ )}{D t} = \frac{\partial ( \ )}{\partial t} + ( \vec{V} \cdot \nabla )( \ ) $
The dot product of the velocity vector, $ \vec{V} $, and the gradient operator, $ \nabla ( \ ) $, provides a convenient notation for the spatial derivative terms appearing in the Cartesian coordinate representation of the material derivative.
Note that the notation $ \vec{V} \cdot \nabla ( \ ) $ represents the operator: $ \vec{V} \cdot \nabla ( \ ) = v_x \frac{\partial ( \ )}{\partial x} + v_y \frac{\partial ( \ )}{\partial y} + v_z \frac{\partial ( \ )}{\partial z} $
Substituting this into the material derivative, we get: $ \frac{D( \ )}{D t} = \frac{\partial ( \ )}{\partial t} + \vec{V} \cdot \nabla ( \ ) $
This compact notation is frequently used in fluid mechanics to express acceleration and other time-dependent properties of fluid motion.