I'm reading a paper where the author introduced the divergence form of the Coriolis force. $$ \omega \times \bf{u} = \bf{u} \cdot \nabla(\omega \times \bf{r}) \ \ \ \ (1) $$ where $ \omega, \bf{u}\ $and$\ \bf{r}$ are the rotating speed vector of the reference frame relative to the absolute inertia frame, the velocity vector and the distance vector. From this the author derives the formula for the Lorentz force where the magnetic field is not constant: $$ \bf{J}\times \bf{B} = -\bf{J}\ \cdot\ \nabla(\bf{B}\ \times \ \bf{r})\ +\ (J \cdot \ \nabla\bf{B})\ \times \ r \ \ \ \ (2) $$ with $ \bf{J},\ \bf{B} $ and $\bf{r}$ are the current density, the magnetic field and the distance vector.

I would like to know how to derive relation (1) and (2).

I'm reading a paper (Ni et al, J Comp Phys (2007)) where the author introduced the divergence form of the Coriolis force. $$ \omega \times \mathbf{u} = \mathbf{u} \cdot \nabla(\omega \times \mathbf{r}) \tag{1} $$ where $ \omega, \mathbf{u}\ $and$\ \mathbf{r}$ are the rotating speed vector of the reference frame relative to the absolute inertia frame, the velocity vector and the distance vector. From this the author derives the formula for the Lorentz force where the magnetic field is not constant: $$ \mathbf{J}\times \mathbf{B} = -\mathbf{J}\ \cdot\ \nabla(\mathbf{B}\ \times \ \mathbf{r})\ +\ (J \cdot \ \nabla\mathbf{B})\ \times \mathbf r\tag{2} $$ with $ \bf{J},\ \bf{B} $ and $\bf{r}$ are the current density, the magnetic field and the distance vector.

I would like to know how to derive relation (1) and (2).

I'm reading a paper where the author introduced the divergence form of the Coriolis force. $$ \omega \times \bf{u} = \bf{u} \cdot \nabla(\omega \times \bf{r}) \ \ \ \ (1) $$ where $ \omega, \bf{u}\ $and$\ \bf{r}$ are the rotating speed vector of the reference frame relative to the absolute inertia frame, the velocity vector and the distance vector. From this the author derives the formula for the Lorentz force where the magnetic field is not constant: $$ \bf{J}\times \bf{B} = -\bf{J}\ \cdot\ \nabla(\bf{B}\ \times \ \bf{r})\ +\ (J \cdot \ \nabla\bf{B})\ \times \ r \ \ \ \ (2) $$ with $ \bf{J},\ \bf{B} $ and $\bf{r}$ are the current density, the magnetic field and the distance vector.

I would like to know how to derive relation (1) and (2).

Best regards, Roman

I'm reading a paper where the author introduced the divergence form of the Coriolis force. $$ \omega \times \bf{u} = \bf{u} \cdot \nabla(\omega \times \bf{r}) \ \ \ \ (1) $$ where $ \omega, \bf{u}\ $and$\ \bf{r}$ are the rotating speed vector of the reference frame relative to the absolute inertia frame, the velocity vector and the distance vector. From this the author derives the formula for the Lorentz force where the magnetic field is not constant: $$ \bf{J}\times \bf{B} = -\bf{J}\ \cdot\ \nabla(\bf{B}\ \times \ \bf{r})\ +\ (J \cdot \ \nabla\bf{B})\ \times \ r \ \ \ \ (2) $$ with $ \bf{J},\ \bf{B} $ and $\bf{r}$ are the current density, the magnetic field and the distance vector.

I would like to know how to derive relation (1) and (2).