Gradient theorem by the divergence theorem

[advphys]

Hi to all

Homework Statement

∫∫∫∇ψdv = ∫∫ψ ds
over R over S

R is the region closed by a surface S

here dv and ψ are given as scalars and ds is given as a vector quantitiy.
and questions asks for establishing the gradient theorem by appliying the divergence theorem to each component

Homework Equations

Divergence theorem
∫∫∫∇.Fdv=∫∫F.ds

The Attempt at a Solution

i tried writing ψ as some components or some functions of F. i actually tried lots of thing like writing ds or ψ or F in open forms.
but probably i do not understand what it means by "to each component"
and i couldn't find the way that i should approach to the question.

this homework is due tomorrow, so i would really appreciate any help.

Thanks a lot.

 

[BruceW]

∫∫∫∇ψdv = ∫∫ψ ds
over R over S

R is the region closed by a surface S

here dv and ψ are given as scalars and ds is given as a vector quantitiy.
and questions asks for establishing the gradient theorem by appliying the divergence theorem to each component

I don't think this is right. ds is not making a dot product with anything. Also, integrating a gradient over a volume will not get you anything useful.

Divergence theorem
∫∫∫∇.Fdv=∫∫F.ds

This is the key. Start from this. Now, this equation involves the vector F. But you can also 'interpret' this as an equation involving three scalars, which are each of the components of F. Start with the simple case, where only one of the components of F is non-zero. Now, see what you get. And think of the equation for the gradient theorem, try to get to that. And make your equation as simple as you want, what should you make as the Gaussian surface, and what should F depend on? (it is your choice, remember)

 

[advphys]

Hmm, ok.

I choose as F= Fx i + 0j+0k
∇.F=Fx
then the right side becomes also integration of (Fx ds sub x)
so, for the x component i think i can say
∇ψ=Fx
but here ∇ψ is a vector quantitiy but Fx is scalar now. How could it be?

 

[BruceW]

hold on. You chose F=Fx i
That is good. So from here, what is ∇.F ?
Right-hand side is correct, it is Fx ds (where ds is perpendicular to x direction).
And the idea is that Fx=ψ.

 

[unfisico]

i'm a bit late...english isn't my native languaje so please excuse my gramatical errors... the man that replied you made a mistake. the identity is well written as you wrote it. there is nothing wrong with the vector quantity dS, that stands for N.ds being N the normal of the surface. This thing that you have there is the gradient theorem generalization for 3d manifolds. It is prooven quite like the divergence theorem using the definition of gradient that you will find in the text I'm providing you. If you speak spanish, this: http://www.esi2.us.es/DFA/CEMI/Teoria/Tema1.pdf could help. search for "teorema del gradiente" and you will finde the gradient theorem. Search for "Definición intrínseca de gradiente" and you will find the definition of gradient that you must use for the proof. good luck.