Question

fan will medal and.






Hey! I have a problem with Curl/Div/Gradient !

http://screencast.com/t/MxofUMGMPgeZ

Who can help me prove this thing ?

What is Gradient(Phi) dot F ???

Answer 1

here is how this "proof" is laid out in Boas, slightly different symbology but it's good to be flexible.

https://i.gyazo.com/e0637b192c1c2410676216a9a5475eb5.png

this might not be too legible so i gave the link first :)

Answer 2

Or if you wanna do it long hand in Cartesian....

\(div (\phi \mathbf{F}) = <\partial_x, \partial_y, \partial_z>\cdot <\phi F_x,\phi F_y,\phi F_z > \)

\(= \partial_x(\phi F_x) + \partial_y(\phi F_y) + \partial_z(\phi F_z) \)

and then you product-rule each term

\(= (\partial_x \phi) ~ F_x + \phi ~ \partial_x F_x + (\partial_y \phi) ~ F_y + \phi ~ \partial_y F_y + (\partial_z \phi) ~ F_z + \phi ~ \partial_z F_z \)

Collecting terms:

\(= (\partial_x \phi) ~ F_x + (\partial_y \phi) ~ F_y + (\partial_z \phi) ~ F_z + \phi ( \partial_x F_x + \partial_y F_y + \partial_z F_z) \)

putting back into dot products

\(= <\partial_x \phi , \partial_y \phi , \partial_z \phi > \cdot < F_x , F_y , F_z >+ \phi ( <\partial_x , \partial_y , \partial_z> \cdot < F_x + F_y + F_z>) \)

\(= \nabla ( \phi ) \cdot \mathbf{F} + \phi \nabla \cdot \mathbf{F}\)

\(= \phi div \mathbf{F} + grad ( \phi ) \cdot \mathbf{F} \)

Quite messy