- #1

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What is the Surface Area vector form in exterior algebra ,I mean by that the Surface Area vector as an exterior form in 3D , just like the volume form .

THANKS

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- Thread starter mikeeey
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- #1

- 57

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What is the Surface Area vector form in exterior algebra ,I mean by that the Surface Area vector as an exterior form in 3D , just like the volume form .

THANKS

- #2

Matterwave

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- #3

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this is why i want to know ,

in mechanics the stress distribution formula is [itex] F^i = \sigma^\ij dA_j [/itex]

where F is the force vector and (Sigma ) is the mechanical second order stree tensor and A is the Area vector

while in exterior algebra it's written like this [itex] F^i = T^i_jk dx^j\wedgedx^k [/itex]

where T is a third order tensor , when using calculus e.g. co-variant derivative , sigma with give 2 christoffel symbols while the T will give 3 christoffel symbols

- #4

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[itex] F^i =\sigma^ij dA_j [/itex]

[itex] F^i= B^i_j_k dx^j \wedge dx^k [/itex]

[itex] F^i= B^i_j_k dx^j \wedge dx^k [/itex]

- #5

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In the cross-product, the oriented parallelogram formed from the factors is more fundamental than the vector perpendicular to that parallelogram.

From a tensor algebra viewpoint, to get a vector from the oriented parallelogram,

one has to use the Hodge-dual (often symbolized by *), which involves the [itex]\epsilon_{ijk}[/itex] symbol.

- #6

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you mean [itex] dA_i = \epsilon_ijk dx^j \wedge dx^k [/itex]

- #7

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[itex] dA^i = \epsilon _{ijk} dx^j \wedge dx^k [/itex]

- #8

Matterwave

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[itex] dA^i = \epsilon _{ijk} dx^j \wedge dx^k [/itex]

Yes, it's basically this, but you might have some normalization factors in there, I'm not quite sure.

EDIT: Oh, and in your formula the i has moved from lower index to upper index, so you have to raise the index in there somewhere. :)

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