Gradient theorem by the divergence theorem

In summary, this person is trying to solve a problem involving the divergence and gradient theorem, but is having difficulty understanding what is required.
  • #1
advphys
17
0
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.
 
Physics news on Phys.org
  • #2
advphys said:
∫∫∫∇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.

advphys said:
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)
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 

1. What is the gradient theorem?

The gradient theorem is a fundamental theorem in vector calculus that relates the line integral of a scalar field to the gradient of that field. It is also known as the fundamental theorem of calculus for line integrals.

2. What is the divergence theorem?

The divergence theorem, also known as Gauss's theorem, is a fundamental theorem in vector calculus that relates the surface integral of a vector field to the volume integral of the divergence of that field. It is a higher-dimensional analog of the fundamental theorem of calculus for line integrals.

3. How is the gradient theorem related to the divergence theorem?

The gradient theorem and the divergence theorem are closely related as they both involve integrals and derivatives of vector fields. The gradient theorem is a special case of the divergence theorem, where the vector field is the gradient of a scalar field. This means that the gradient theorem can be derived from the divergence theorem.

4. What are some practical applications of the gradient theorem and the divergence theorem?

The gradient theorem and the divergence theorem have many practical applications in physics and engineering. For example, they are used to calculate electric and magnetic fields, fluid flow, heat transfer, and many other physical phenomena. They are also used in computer graphics and image processing for gradient and divergence-based algorithms.

5. Are there any limitations to the gradient theorem and the divergence theorem?

Like any mathematical theorem, the gradient theorem and the divergence theorem have their own limitations. They are only valid for smooth vector fields and well-behaved regions of integration. In addition, they may not hold in higher dimensions or in non-Euclidean spaces. It is important to carefully consider the assumptions and limitations when applying these theorems in a particular problem.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
741
  • Calculus and Beyond Homework Help
Replies
7
Views
389
  • Calculus and Beyond Homework Help
Replies
20
Views
3K
  • Calculus and Beyond Homework Help
Replies
5
Views
2K
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
4
Views
914
  • Calculus and Beyond Homework Help
Replies
8
Views
348
  • Calculus and Beyond Homework Help
Replies
6
Views
2K
  • Calculus and Beyond Homework Help
Replies
8
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
930
Back
Top