Transform rule for (1,2) tensor

In summary: The transformation rule for tensors does not require that additional term. In summary, the transformation rule for a (1,2) tensor is T^a{}_{bc} = \bar T^d{}_{ef}\frac{\partial x^a}{\partial \bar x^d}\frac{\partial \bar x^e}{\partial x^b}\frac{\partial \bar x^f}{\partial x^c}. The added term, \frac{\partial x^i}{\partial x^k}\frac{\partial^2 \bar x^m}{\partial x^j \partial x^k}, is only necessary when dealing with Christoffel symbols of the second kind, which have an additional term due
  • #1
franznietzsche
1,504
6
What is the transform rule for a (1,2) tensor? Is it:

[tex]

T^a{}_{bc} = \bar T^d{}_{ef}\frac{\partial x^a}{\partial \bar x^d}\frac{\partial \bar x^e}{\partial x^b}\frac{\partial \bar x^f}{\partial x^c}

[/tex]

or is there an added term like in the transform rule for Christoffel symbols of the second kind?
 
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  • #2
Originally posted by franznietzsche

or is there an added term like in the transform rule for Christoffel symbols of the second kind?
this is correct. you only need the Christoffel symbols if you are taking a derivative.
 
  • #3


Thank you.

Originally posted by lethe
this is correct. you only need the Christoffel symbols if you are taking a derivative.

i know, but that transofrm is the same as the one for the christoffel symbols except the christoffel symbol's transformation rule has the added term:
[tex]
\frac{\partial x^i}{\partial x^k}\frac{\partial^2 \bar x^m}{\partial x^j \partial x^k}
[/tex]

That is what i was referring to.
 
  • #4


Originally posted by franznietzsche
Thank you.



i know, but that transofrm is the same as the one for the christoffel symbols except the christoffel symbol's transformation rule has the added term:
[tex]
\frac{\partial x^i}{\partial x^k}\frac{\partial^2 \bar x^m}{\partial x^j \partial x^k}
[/tex]

That is what i was referring to.
yeah, the Christoffel symbols have an additional term, because they involve taking a derivative.

since you are not, then you do not need the extra term, and the equation you have in the first post is correct.
 
  • #5
tensors always transform always like you wrote [tex]T^a{}_{bc} = \bar T^d{}_{ef}\frac{\partial x^a}{\partial \bar x^d}\frac{\partial \bar x^e}{\partial x^b}\frac{\partial \bar x^f}{\partial x^c}[/tex]
(by definition)
When you take a normal derivative from a tensor, you don't become a tensor. This is a problem for making diff equations with tensors. Therefor we define a new derivative (covariant derivative)
We becomes this by putting a second term (connection coefficients) by ten partial derivative. In general relativity we take a connection coëfficient we have derived from the metric. This is the Christoffel connection(are symbols)
 
  • #6
The Christoffel symbols have that added term specifically because the Christoffel symbols are not tensors.
 

1. What is the transform rule for a (1,2) tensor?

The transform rule for a (1,2) tensor is a mathematical equation that describes how the components of the tensor change when the coordinate system is transformed. It is used to determine how a tensor behaves under different coordinate systems, making it a useful tool in various fields of science and engineering.

2. What does the (1,2) notation mean in a tensor?

The (1,2) notation in a tensor refers to the number of contravariant and covariant indices, respectively. In a (1,2) tensor, there is one contravariant index and two covariant indices, indicating that the tensor has three components.

3. How is the transform rule for a (1,2) tensor derived?

The transform rule for a (1,2) tensor is derived using the principles of tensor calculus and linear algebra. It involves transforming the basis vectors and then applying the transformation to the components of the tensor using a matrix multiplication. This results in a set of equations that describe the transform rule for the tensor.

4. What is the significance of the transform rule for a (1,2) tensor?

The transform rule for a (1,2) tensor is significant because it allows us to work with tensors in different coordinate systems, making it easier to solve problems and analyze data. It also helps us understand how physical quantities behave under different transformations, providing insights into the underlying properties of the system.

5. Can the transform rule for a (1,2) tensor be applied to tensors of other ranks?

Yes, the transform rule for a (1,2) tensor can be applied to tensors of any rank. However, the notation and equations may differ depending on the rank of the tensor. For example, a (1,2) tensor has three components, while a (2,1) tensor has six components, and their transform rules will reflect this difference.

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