Correct Expression of the Material Derivative in Continuum Physics

The material derivative is important in continuum physics. This Letter shows that the expression \(\frac{d }{dt}=\frac{\partial }{\partial t}+(\bm v\cdot \bm \nabla)\), used in most literature and textbooks, is incorrect. The correct expression $ \frac{d (:)}{dt}=\frac{\partial }{\partial t}(:)+\bm v\cdot [\bm \nabla (:)]$ is formulated. The solution existence condition of Navier-Stokes equation has been proposed from its form-solution, the conclusion is that "\emph{The Navier-Stokes equation has a solution if and only if the determinant of flow velocity gradient is not zero, namely \(\det (\bm \nabla \bm v)\neq 0\).}"