Micromechanics of composites/Proof 4

Question

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Let   be a surface. Let   be the position vector of points on the surface and let   be a vector field that are defined on  . If

 

show that

 

Proof

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If we assume a Cartesian basis, we can write the given relation in index notation as

 

where   is the Levi-Civita (permutation) symbol. Since   is does not depend upon the position we can write

 

Define

 

Then,

 

Expanding, we get

 

Expanding further, we get three equations

 

or,

 

or,

 

Therefore,

 

Also, by symmetry,

 

Therefore we may write,

 

Reverting back to direct tensor notation, we get