Ñ.(fa)=fÑ.a+a.Ñf

By Dan Coomber on Wednesday, April 17, 2002 - 03:07 pm:

Can anyone help with proving the following vector identities? I have found many texts stating them, but none with proofs:

With suitably smooth scalar field f and suitably smooth vector fields a and b:

Ñ.(fa)=fÑ.a+a.Ñf

and

Ñ.(a×b)=b.(Ñ×a)-a.(Ñ×b)

Many thanks...

By David Loeffler on Wednesday, April 17, 2002 - 09:52 pm:

Use suffix notation. Write a={a_i}, where i=1, 2, 3. Then

Ñ.(fa)=
å
i
¶/¶x_i(fa_i)

=
å
i
(f¶a_i/¶x_i+a_i¶f/¶x_i)

=fÑ.a+a.(Ñf)

The curl one is similar.

David