Here is the all identities : http://en.wikipedia.org/wiki/Vector_calculus_identities
I need help concerning vector functions and indexing notations.
Let $\overrightarrow{a}$ be a (smooth) vector field and $\varphi$ be a (smooth) scalar function. Show $$ \overrightarrow {\nabla }\cdot \left( \varphi\,\overrightarrow {a}\right) = \varphi \left( \overrightarrow {\nabla }\cdot \overrightarrow {a}\right) +\overrightarrow {a} \cdot \overrightarrow {\nabla }\varphi.$$
I have to use this notation to prove this, but how?
I don't really understand.
My second identity is ;
$$ \overrightarrow {\nabla }\times \left( \phi \cdot \overrightarrow {a}\right)
$$
Best Answer
Here's what's happening in $\mathbb{R}^3$ with rectangular coordinates. You can tweak as needed.
Let $g(x,y,z)$ be a smooth scalar function and $\mathbf{F}(x,y,z)=(F_1(x,y,z),F_2(x,y,z),F_3(x,y,z))$ be a smooth vector field. Then \begin{align} \nabla \cdot (g\,\mathbf{F})&=\nabla\cdot((gF_1,gF_2,gF_3))\\ &=(gF_1)_x+(gF_2)_y+(gF_3)_z\\ &=g_xF_1+g(F_1)_x+g_yF_2+g(F_2)_y+g_zF_3+g(F_3)_z, \end{align} while \begin{align} \nabla g\cdot \mathbf{F}&=g_x F_1+g_y F_2+g_z F_3,\\ g\,(\nabla\cdot \mathbf{F})&=g\,((F_1)_x+(F_2)_y+(F_3)_z)=g(F_1)_x+g(F_2)_y+g(F_3)_z. \end{align} Adding these last two yields the first.