A confusing vector field differential In my notes on theoretical mechanics, I wrote that my professor stated this vector identity:
$$\mathrm{d}\mathbf{P}(\mathbf{r})=[\nabla\cdot\mathbf{P}(\mathbf{r})] \mathbf{dr} + [\nabla\times\mathbf{P}(\mathbf{r})]\times\mathbf{dr}$$
Here, $\mathbf{P}(\mathbf{r})$ stands for the vector field depending only on the position vector $\mathbf{r}$. In other words, it's a vector-valued function of a vector in $\mathbb{R}^3$.
The thing is, I didn't manage to write down what the importance was, although I think it had something to do with separating parallel and perpendicular differential vectors. 
I tried to find something like this on the Internet and I also tried to show it, but I'm afraid I can't find an elegant way to show this and I don't think I'll gain anything by mechanically grinding it out.
What is the significance of this identity? Is there a simple way to prove it?
 A: Curious formula, I'm not sure I've seen this before. That said, let's have a go at it.
Let $\vec{P} = \langle \ A, \ B, \ C  \ \rangle$ and suppose $\vec{dr} = \langle dx,dy,dz \rangle$. The total differential of the vector field is:
\begin{align} 
d\vec{P} 
&= \langle \ dA, \ dB, \ dC  \ \rangle  \\ 
&= \langle \ A_xdx+A_ydy+A_zdz, \ B_xdx+B_ydy+B_zdz, \ C_xdx+C_ydy+C_zdz  \ \rangle  \\ 
&= (A_x+B_y+C_z ) \langle dx, dy, dz \rangle -\langle B_ydx+C_zdx,A_xdy+C_zdy,A_xdz+B_ydz \rangle 
\\ 
& \qquad + \langle \ A_ydy+A_zdz, \ B_xdx+B_zdz, \ C_xdx+C_ydy  \ \rangle   \\ 
&= (\nabla \cdot \vec{P}) \vec{dr} + \langle  A_ydy+A_zdz-B_ydx-C_zdx,  B_xdx+B_zdz-A_xdy-C_zdy, \ \ \ \ \ \ \qquad C_xdx+C_ydy-A_xdz-B_ydz  \rangle   \\
\end{align}
Well, I must go eat supper, but I think we see where this is going. It was delicious, and now we continue, let's study the curl term
\begin{align} 
[\nabla \times \vec{P}] \times \vec{dr}
&= \langle C_y-B_z, A_z-C_x, B_x-A_y\rangle \times \langle dx,dy,dz \rangle \\ 
&= \langle (A_z-C_x)dz-(B_x-A_y)dy, \ (B_x-A_y)dx-(C_y-B_z)dz, \\ &\qquad (C_y-B_z)dy-(A_z-C_x)dx \rangle  
\end{align}
Well, if I made an error, I don't see it at present. The calculation above does not support the identity claimed. It has the right signs, but, the differentials and partial derivatives need to be swapped on six terms. I hope someone else sheds further light on this.
