I want to verify if this vector identity is true and if so is there a simple proof of it? $\textbf{u}$ is a vector and $P$ is a scalar field, namely the pressure.

\begin{equation} (\mathbf{\nabla} \cdot \textbf{u})P = (\mathbf{\nabla} P) \cdot \textbf{u} \end{equation}

  • $\begingroup$ It does not seem this identity holds for any vector field $\textbf{u}$ and scalar $P$. $\endgroup$ – Reza Jan 10 '18 at 22:52

In Cartesian coordinates and using index notation one finds \begin{equation} (\nabla \cdot u) P = (\partial_i u_i) P \end{equation} but on the other hand \begin{equation} (\nabla P)\cdot u = (\partial_i P) u_i \end{equation} In general these are not equal.

Notice that \begin{equation} (\nabla \cdot u) P = (\partial_i u_i) P = \partial_i (u_i P) - (\partial_i P) u_i = \nabla\cdot(P u) - u\cdot\nabla P \end{equation} Even if $\nabla\cdot(P u) = 0$, there would be a sign difference with your statement.

  • $\begingroup$ Thank you, I see the problem. $\endgroup$ – hahahasan Jan 10 '18 at 23:21

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