Divergence theory equalities

In summary, these equations involve the use of the divergence theorem and index notation to solve for the given functions and variables.
  • #1
boyboy400
9
0

Homework Statement



So I got three things to figure out:

1- ∫Curl u dV=∫u χ n dS
2- ∫div Tu dV=∫TT n. udS
3- ∫div θu dV=∫n.θu udS

where
n defines the outward normal to the boundary S
θ is a smooth scalar-valued function
u is a smooth vector-valued function
T is a smooth tensor-valued function

Homework Equations


The Attempt at a Solution


1- Let Tijijkuk
and ∫Tij,jdV=∫TijnjdS
Substituting the first one into the integral one (second one) we get the indices form of what we want. So it's solved.

2- ∫(∂Tij/∂xiUj)dV=∫TijUjUidS
but from here I don't know where to go!

3- I guess if the second one is solved the last one would be easy.

PS. In case these relations have a special name or there is a keyword I can google and find my answers I really appreciate if you can tell me about. Also if there is a book that has the solution please let me know about it. Thank you so much everyone

PS2. Well using the definition of divergence theorem and index notation, I managed to write something...it seems kind of clear but I'm not sure about playing around with the orders and indices especially for the second one where Transpose[T] has to be made at the right hand side like I don't know how to do this ... so hopefully the TA will not be picky this time :D
 
Last edited:
Physics news on Phys.org
  • #2
1- Using the divergence theorem: ∫Curl u dV=∫div(T)dV=∫Tn.udS where T is a smooth tensor-valued function and n is the outward normal to the boundary S.2- Using the definition of curl and index notation: ∫(∂Tij/∂xiUj)dV=∫TijUjUidS where U is a smooth vector-valued function.3- Using the definition of divergence and index notation:∫div θu dV=∫n.θu udS where θ is a smooth scalar-valued function and n is the outward normal to the boundary S.
 

What is Divergence Theory?

Divergence theory is a scientific theory that explains how species evolve and diverge from a common ancestor over time. It suggests that differences in traits and characteristics among species are due to natural selection and adaptation to different environments.

How does Divergence Theory explain species diversity?

Divergence theory explains species diversity by proposing that as species adapt to different environments and undergo natural selection, they develop unique traits and characteristics that make them better suited for survival in their specific habitat. Over time, these differences can lead to the formation of new species.

What evidence supports Divergence Theory?

There is a wealth of evidence that supports Divergence Theory, including fossil records, genetic studies, and observations of species in their natural habitats. Fossils show a progression of species over time, with gradual changes in traits and characteristics. Genetic studies reveal similarities and differences among species, providing insight into their evolutionary relationships. Observations of species in their natural habitats also demonstrate how natural selection and adaptation can lead to divergence.

How does Divergence Theory differ from other theories of evolution?

Divergence Theory differs from other theories of evolution, such as convergent evolution or punctuated equilibrium, in its emphasis on gradual change and the accumulation of small differences over time. It also focuses on the role of natural selection and adaptation in driving species divergence.

Are there any criticisms of Divergence Theory?

Like any scientific theory, Divergence Theory has faced some criticisms. Some argue that it does not fully explain the rapid diversification of species during certain periods in Earth's history. Others suggest that it may oversimplify the complex process of evolution and the factors that contribute to species diversity. However, Divergence Theory remains a widely accepted and extensively studied theory in the scientific community.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
688
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
1K
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
664
  • Calculus and Beyond Homework Help
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
1
Views
749
  • Calculus and Beyond Homework Help
Replies
22
Views
2K
  • Calculus and Beyond Homework Help
Replies
1
Views
601
Back
Top