Vector derivative identity

[quantumfoam]

Hello I would like to know if div(f(grad(f)))=f(grad^2)f, where f is some scalar function and (grad^2) is the Laplacian. I am having trouble seeing the difference.

 

[quantumfoam]

I would also like to say that this is no homework question. I love doing mathematics during my spare time

 

[HallsofIvy]

Did you try working it out for a general function f? grad f is the vector \(\displaystyle <f_x, f_y, f_z>\) so the scalar product f grad f is equal to \(\displaystyle <ff_x, ff_y, ff_z>\) and then dif (f grad f) is \(\displaystyle (ff_x)_x+ (ff_y)_y+ (ff_z)_z\). By the product rule, \(\displaystyle (ff_x)_x= (f_x)f_x+ f(f_{xx})\) and similarly for y and z. So \(\displaystyle dif (f grad f)= (f_x)^2+ ff_{xx}+ (f_y)^2+ ff_{yy}+ (f_z)^2+ ff_{zz}\).

On the other side, \(\displaystyle grad^2 f= f_{xx}+ f_{yy}+ f_{zz}\) and then \(\displaystyle f(grad^2 f)= ff_{xx}+ ff_yy+ ff_{zz}\).