# Vector Calculus Mysterious Identity

1. Jul 28, 2009

### sokrates

Hi, I am working on a topic called "Generalized Magnetization Dynamics" and the following identity is claimed by the authors. I would be very much grateful if you could give me an expert view on the subject:

Given :

$$\textbf{m} \times \nabla_\Sigma f = \textbf{m} \times \frac{\partial f}{\partial \textbf {m}}$$

authors claim:

$$\nabla_\Sigma f = - \textbf{m} \times \textbf{m} \times \frac{\partial f}{\partial \textbf{m}}$$

holds for any differentiable function f (m). Here m is a unit vector so it's magnitude is unity. And this is the only requirement from the physics point of view. Gradient with respect to Sigma indicates that the gradient operator acts on the scalar field on the unit sphere Sigma.

But performing the BAC - CAB rule on the second equation tells me that this identity is only true when:

$$\textbf{m} \cdot \frac{\partial f}{\partial \textbf{m}} = 0$$

which is not obvious to me.

Am I missing something?

The book is Nonlinear Magnetization Dynamics in Nanosystems - Page : 49.
(You can view the page in amazon by searching '49' inside the book if you have an account)