nabla acting on products

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
$\nabla(\vec{u}\cdot\vec{v})\,=\,(\vec{v}\cdot\nabla)\vec{u}+(\vec{u}\cdot% \nabla)\vec{v}+\vec{v}\!\times\!(\nabla\!\times\!\vec{u})+\vec{u}\!\times\!(% \nabla\!\times\!\vec{v})$
$\nabla(\vec{u}\cdot\vec{v})=(\nabla\vec{u})\cdot\vec{v}+(\nabla\vec{v})\cdot% \vec{u}$

• 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. 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. 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. 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 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.

 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