Can you explain dual tensors and tensor densities with a simple example?

[aditya23456]

Can anyone explain dual tensor and complex tensor with a simple situation.And even How tensor density is related to transformation of axis..THANKS IN ADVANCE

 

[aditya23456]

Is my question valid??

 

[Matterwave]

I think you would have to explain the context.

Dual usually means "corresponds with". For the case of a vector, the dual would be a one-form because one-forms are linear functions of vectors to the reals, and vice versa. For a general n-form, "dual" can also sometimes be used to mean the "Hodge dual" (completely unrelated with the above "dual") which gives you a (m-n)-form (m being the dimension of the manifold).

I don't know what the "dual" of a general tensor would be generically.

Complex tensor should just mean a linear map of one-forms and/or vectors into the complex numbers. A complex vector, for example, would simply be a vector over the complex field instead of the real field (essentially, the components can be complex).

I have no idea what you mean by "tensor density is related to transformation of axis". A tensor density is usually just a tensor multiplied by a factor sqrt(g). Nothing really special about them, they just have a slightly different covariant derivative than a real tensor and they transform somewhat differently, and are mostly used out of convenience rather than necessity (as far as I have experience with them).

 

[aditya23456]

is there any link where all this is explained WITH EXAMPLE OF A SITUATION..i guess example will give a better insight in its applications

 

[Muphrid]

I have no idea what you're talking about when you refer to "complex tensors."

Sometimes, tensors are linear operators. For example, the metric tensor is used to act on a vector to convert it from a tangent basis to a cotangent basis. These tensors do not, to my knowledge, have duals.

Other tensors represent physical objects in a vector space. They represent directed line segments (vectors), planes, volumes, and so on. For these objects, the dual tensor represents the subspace that complements the object. For example, in 3d, the dual tensor to a vector represents a plane (vector + plane fully spans 3d space).

Tensor densities are a bit of a cheat when talking about integrals of tensors with respect to volumes. Think about an integral when you change variables from dx to du = f' dx. When you substitute for dx in that integral, you end up with a term of 1/f' du. Tensor densities roll in that factor of 1/f' to reduce the amount of bookkeeping you have to do.