nabla acting on products

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

• Gradient of a product function
  $\nabla (fg)=(\nabla f)g+(\nabla g)f$

• Divergence of a scalar-multiplied vector
  $\nabla \cdot (f\overrightarrow{u})=(\nabla f)\cdot \overrightarrow{u}+(\nabla \cdot \overrightarrow{u})f$

• Curl of a scalar-multiplied vector
  $\nabla \times (f\overrightarrow{u})=(\nabla f)\times \overrightarrow{u}+(\nabla \times \overrightarrow{u})f$

• Divergence of a vector product
  $\nabla \cdot (\overrightarrow{u}\times \overrightarrow{v})=(\nabla \times \overrightarrow{u})\cdot \overrightarrow{v}-(\nabla \times \overrightarrow{v})\cdot \overrightarrow{u}$

• Curl of a vector product
  $\nabla \times (\overrightarrow{u}\times \overrightarrow{v})=(\overrightarrow{v}\cdot \nabla )\overrightarrow{u}-(\overrightarrow{u}\cdot \nabla )\overrightarrow{v}-(\nabla \cdot \overrightarrow{u})\overrightarrow{v}+(\nabla \cdot \overrightarrow{v})\overrightarrow{u}$

• Gradient of a scalar product
  $\nabla (\overrightarrow{u}\cdot \overrightarrow{v})=(\overrightarrow{v}\cdot \nabla )\overrightarrow{u}+(\overrightarrow{u}\cdot \nabla )\overrightarrow{v}+\overrightarrow{v}\times (\nabla \times \overrightarrow{u})+\overrightarrow{u}\times (\nabla \times \overrightarrow{v})$
  or, using dyads,
  $\nabla (\overrightarrow{u}\cdot \overrightarrow{v})=(\nabla \overrightarrow{u})\cdot \overrightarrow{v}+(\nabla \overrightarrow{v})\cdot \overrightarrow{u}$

• Gradient of a vector product
  $\nabla (\overrightarrow{u}\times \overrightarrow{v})=(\nabla \overrightarrow{u})\times \overrightarrow{v}-(\nabla \overrightarrow{v})\times \overrightarrow{u}$

• Divergence of a dyad product
  $\nabla \cdot (\overrightarrow{u}\overrightarrow{v})=(\nabla \cdot \overrightarrow{u})\overrightarrow{v}+\overrightarrow{u}\cdot \nabla \overrightarrow{v}$

• Curl of a dyad product
  $\nabla \times (\overrightarrow{u}\overrightarrow{v})=(\nabla \times \overrightarrow{u})\overrightarrow{v}-\overrightarrow{u}\times \nabla \overrightarrow{v}$

Explanations

1. $\overrightarrow{v}\cdot \nabla$ 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 $\overrightarrow{w}$ is defined as the dyad $\nabla \overrightarrow{w}:=\overrightarrow{i}\frac{\partial \overrightarrow{w}}{\partial x}+\overrightarrow{j}\frac{\partial \overrightarrow{w}}{\partial y}+\overrightarrow{k}\frac{\partial \overrightarrow{w}}{\partial z}$.

3. The divergence and the curl of a dyad product are defined by the equation
   $\nabla *(\overrightarrow{u}\overrightarrow{v}):=\overrightarrow{i}*\frac{\partial (\overrightarrow{u}\overrightarrow{v})}{\partial x}+\overrightarrow{j}*\frac{\partial (\overrightarrow{u}\overrightarrow{v})}{\partial y}+\overrightarrow{k}*\frac{\partial (\overrightarrow{u}\overrightarrow{v})}{\partial z}$,  where the asterisks are dots or crosses and the partial derivatives of the dyad product the expression $\frac{\partial (\overrightarrow{u}\overrightarrow{v})}{\partial x}=\frac{\partial \overrightarrow{u}}{\partial x}\overrightarrow{v}+\overrightarrow{u}\frac{\partial \overrightarrow{v}}{\partial x}$  and so on.

Title	nabla acting on products
Canonical name	NablaActingOnProducts
Date of creation	2013-03-22 15:27:05
Last modified on	2013-03-22 15:27:05
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	11
Author	pahio (2872)
Entry type	Topic
Classification	msc 26B12
Classification	msc 26B10
Related topic	Nabla
Related topic	NablaNabla
Defines	gradient of vector
Defines	divergence of dyad product
Defines	curl of dyad product