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E&M proof for a integralnot sure whats it called

  • #1
101
0

Homework Statement


Δ<--- the gradient X<---cross product
(o∫) closed integral

((ΔT)Xda = (o∫)-Tdl

btw what is the realtion ship above called?

Homework Equations


da= axdydz +aydxdz +azdxdy
dl=axdx + aydy + azdy

The Attempt at a Solution


not sure where to go on from here :/ im kinda lost
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
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Last edited by a moderator:
  • #3
213
8
From what I can make out of the question you are supposed to prove that

[tex]\int \nabla T \times d\bold a = \oint T d\bold I[/tex]

where [tex] d\bold a = (dydz,dxdz,dxdy) , d\bold I=(dx,dy,dz)[/tex]

Am I right
 
  • #4
101
0
yes, im not sure how to go about it.
im suppose to show that both of them are equal to each other
[tex]
\int \nabla T \times d\bold a = - \oint T d\bold I
[/tex]
 
  • #5
213
8
its a bit tedious if you don't use the Levi-Civita tensor, but letting dxdy=-dydx (I know it's a bit weird but it's the only way this thing works) the LHS =

[tex] \int \int \frac{1}{2} \epsilon_{ijk} \epsilon_{klm} \partial_{j} T dx^l dx^m [/tex]

[tex]= \int \int \frac{1}{2} (\delta_{il} \delta_{jm} - \delta_{im} \delta_{jl}) \partial_{j} T dx^l dx^m[/tex]

[tex]= \int \int \frac{1}{2} (\partial_{m} T dx^i dx^m - \partial_{l} T dx^l dx^i)[/tex]

[tex]= \int \int \frac{1}{2} (-\partial_{m} T dx^m dx^i - \partial_{l} T dx^l dx^i) = - \oint T dx^i [/tex]

where we used the contracted epsilon identity and changed the last integral into a closed one because going from integrating with respect to an area to a line integral the boundaries change so that the integral becomes closed

Anyway hope this helped the way to do it using standard vector operations is to rewrite [tex] d\bold a = (dydz,dxdz,dxdy) [/tex] as [tex] d\bold a = \frac{1}{2} (dydz -dzdy,dxdz-dzdx,dxdy-dydx) [/tex] which makes sense because the area between two vectors is [tex]\left|\bold a \times \bold b \right| [/tex]
 
Last edited:
  • #6
101
0
thanks for the help, but i have not learnt it the way you have shown above so its really hard for me to understand it. :/

is there a simpler way?
 
  • #7
213
8
well do you know what the cross product between two vectors is
 

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