Surface Area Vector in Exterior Algebra 3D

In summary, the Surface Area vector in exterior algebra is a 2-form that gives a definition of an n-dimensional volume.
  • #1
57
0
Hello every one .
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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  • #2
Wouldn't that just be a 2-form? A n-form gives a definition of an n-dimensional volume, so an area should be associated with a 2-form.
 
  • #3
yes its a 2-forms , but 2-form is a co-variant second order tensor , but here the surface area is a vector ,
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
[itex] F^i =\sigma^ij dA_j [/itex]
[itex] F^i= B^i_j_k dx^j \wedge dx^k [/itex]
 
  • #5
The "surface area element" can be thought of as a "[3D-]vector" only in 3-D.
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
you mean [itex] dA_i = \epsilon_ijk dx^j \wedge dx^k [/itex]
 
  • #7
[itex] dA^i = \epsilon _{ijk} dx^j \wedge dx^k [/itex]
 
  • #8
mikeeey said:
[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. :)
 

What is the "Surface Area Vector" in Exterior Algebra 3D?

The "Surface Area Vector" in Exterior Algebra 3D is a mathematical concept used to calculate the surface area of a three-dimensional object. It is represented by a vector, which has both magnitude and direction.

How is the Surface Area Vector calculated?

The Surface Area Vector is calculated by taking the cross product of two vectors that lie on the surface of the object. These vectors are known as tangent vectors and are perpendicular to each other. The magnitude of the Surface Area Vector is equal to the area of the parallelogram formed by the two tangent vectors.

What is the significance of the Surface Area Vector in Exterior Algebra 3D?

The Surface Area Vector is important in Exterior Algebra 3D because it allows for the calculation of surface area in a more general and efficient way. It also has applications in physics, such as calculating the surface area of a curved object.

Can the Surface Area Vector be used for any three-dimensional object?

Yes, the Surface Area Vector can be used for any three-dimensional object with a well-defined surface. This includes both regular and irregular shapes.

Are there any limitations to using the Surface Area Vector in Exterior Algebra 3D?

One limitation of using the Surface Area Vector is that it only calculates the surface area of an object and not the volume. Additionally, it may not be useful for objects with complex or changing surfaces, such as fluid dynamics systems.

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