Let $f$ $g$ be differentiable scalar fields and $\vec{u}$ $\vec{v}$ differentiable vector fields in some domain of $\mathbb{R}^3$ There are following formulae:

• Gradient of a product function
  $\nabla(fg) = (\nabla f)g+(\nabla g)f$
• Divergence of a scalar-multiplied vector
  $\nabla\cdot(f\vec{u}) = (\nabla f)\cdot\vec{u}+(\nabla\cdot\vec{u})f$
• Curl of a scalar-multiplied vector
  $\nabla\!\times\!(f\vec{u}) = (\nabla f)\times\vec{u}+(\nabla\!\times\!\vec{u})f$
• Divergence of a vector product
  $\nabla\cdot(\vec{u}\!\times\!\vec{v}) = (\nabla\!\times\!\vec{u})\cdot\vec{v}-(\nabla\!\times\!\vec{v})\cdot\vec{u}$
• Curl of a vector product
  $\nabla\!\times\!(\vec{u}\!\times\!\vec{v}) = (\vec{v}\cdot\nabla)\vec{u}-(\vec{u}\cdot\nabla)\vec{v} -(\nabla\cdot\vec{u})\vec{v}+(\nabla\cdot\vec{v})\vec{u}$
• Gradient of a scalar product
  $\nabla(\vec{u}\cdot\vec{v})\, =\, (\vec{v}\cdot\nabla)\vec{u}+(\vec{u}\cdot\nabla)\vec{v} +\vec{v}\!\times\!(\nabla\!\times\!\vec{u}) +\vec{u}\!\times\!(\nabla\!\times\!\vec{v})$ or, using dyads,
  $\nabla(\vec{u}\cdot\vec{v}) = (\nabla\vec{u})\cdot\vec{v}+(\nabla\vec{v})\cdot\vec{u}$
• Gradient of a vector product
  $\nabla(\vec{u}\!\times\!\vec{v}) = (\nabla\vec{u})\!\times\!\vec{v}-(\nabla\vec{v})\!\times\!\vec{u}$
• Divergence of a dyad product
  $\nabla\cdot(\vec{u}\,\vec{v}) = (\nabla\!\cdot\!\vec{u})\,\vec{v}+\vec{u}\cdot\nabla\vec{v}$
• Curl of a dyad product
  $\nabla\!\times\!(\vec{u}\,\vec{v}) = (\nabla\!\times\!\vec{u})\,\vec{v}-\vec{u}\times\!\nabla\vec{v}$

Explanations

1. $\vec{v}\cdot\nabla$ means the operator $v_x\frac{\partial}{\partial x}+v_y\frac{\partial}{\partial y} +v_z\frac{\partial}{\partial z}$
2. The gradient of a vector $\vec{w}$ is defined as the dyad $\nabla\vec{w} := \vec{i}\,\frac{\partial\vec{w}}{\partial x} +\vec{j}\,\frac{\partial\vec{w}}{\partial y} +\vec{k}\,\frac{\partial\vec{w}}{\partial z}$
3. The divergence and the curl of a dyad product are defined by the equation
   $\nabla\!*\!(\vec{u}\vec{v}) := \vec{i}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial x} \!+\!\vec{j}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial y} \!+\!\vec{k}\!*\!\frac{\partial(\vec{u}\vec{v})}{\partial z}$ where the asterisks are dots or crosses and the partial derivatives of the dyad product mean the expression $\frac{\partial(\vec{u}\vec{v})}{\partial x} = \frac{\partial\vec{u}}{\partial x}\vec{v}+ \vec{u}\frac{\partial\vec{v}}{\partial x}$ , and so on.