Difference Between T_{a}^{b} & T^{a}_{b}: (1,1) Tensors

In summary, ##T_{a}^{b}## and ##T^{a}_{b}## are both (1,1) tensors that take in a vector and a dual vector to produce a scalar. The only difference between them is the way they are defined as functions, with ##T_{a}^{b}## taking in a vector as its first argument and ##T^{a}_{b}## taking in a dual vector as its first argument. They can be equivalent in some cases, but in general, they are different tensors.
  • #1
quickAndLucky
34
3
What is the difference between ##{T{_{a}}^{b}}## and ##{T{^{a}}_{b}}## ? Both are (1,1) tensors that eat a vector and a dual to produce a scalar. I understand I could act on one with the metric to raise and lower indecies to arrive at the other but is there a geometric difference between the objects to which these are components?
 
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  • #2
I don't think there's a geometric difference. The difference is that, where ##V## and ##V^*## are the underlying vector space and its dual, ##T_a{}^b## is a function from ##V\times V^*## to ##\mathbb R## and ##T^a{}_b## is a function from ##V^*\times V## to ##\mathbb R##. Or if we think of the tensor as being a function that takes two arguments, ##T_a{}^b## takes a vector as its first argument and ##T^a{}_b## takes a dual vector as its first argument.
 
  • #3
Generally, they are different (1,1) tensors (you could imagine sidestepping #2 by defining the obvious equivalence between functions from ##V\times V^*## and functions from ##V^* \times V##). It is certainly not true that ##T^a_{\phantom ab} \omega_a V^b = T^{\phantom ba}_b \omega_a V^b## except in the case where ##T^a_{\phantom ab} g_{ac}## is symmetric.
 
  • #4
##T_{\ \ \ \nu}^\mu=g_{\nu\xi}T^{\mu\xi}## and

##T^{\ \nu}_\mu=g_{\mu\xi}T^{\xi\nu}##

are different in general. Obviously when T is symmetric

##T^\mu_{\ \ \ \nu}=T^{\ \nu}_\mu## and they can be denoted as ##T^{\nu}_\mu##.
 
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1. What is the difference between Tab and Tab?

The difference lies in the placement of the indices. In Tab, the lower index "a" represents the row and the upper index "b" represents the column. In Tab, the upper index "a" represents the row and the lower index "b" represents the column.

2. How do the indices affect the type of tensor?

The placement of the indices can determine the type of tensor. In Tab, the indices are in a "mixed" form, indicating a (1,1) tensor. In Tab, the indices are in a "pure" form, indicating a (1,1) tensor as well. However, the type of tensor can also depend on the number of indices and their placement in relation to each other.

3. Can a tensor have more than two indices?

Yes, a tensor can have any number of indices. The number of indices determines the order of the tensor. For example, a tensor with three indices is a third-order tensor, also known as a tensor of rank 3.

4. What is the significance of the (1,1) notation for tensors?

The (1,1) notation is used to indicate the type of tensor. The first number represents the number of contravariant indices (upper indices) and the second number represents the number of covariant indices (lower indices). In (1,1) tensors, there is one of each type of index, making it a mixed tensor.

5. How are tensors used in physics and engineering?

Tensors are used to represent physical quantities and their relationships in mathematical models. They are especially useful in fields such as mechanics, electromagnetism, and continuum mechanics. Tensors allow us to describe and analyze complex systems and phenomena by simplifying and organizing the underlying mathematical equations.

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