Cartesian Tensors and some proofs and problems regarding it.

[Raj90]

Homework Statement

I am stuck at this point where I have to prove that the kronecker delta is isotropic tensor.

Homework Equations

δij=δji

The Attempt at a Solution

I know that to prove this I have to show that under coordinate transfor mation it does not change..but it's a bit diff for me to get it right...

 

[TSny]

I have to prove that the kronecker delta is isotropic tensor.


I know that to prove this I have to show that under coordinate transfor mation it does not change.

Hi, Raj90. Welcome to PF.

You don't mean any coordinate transformation, right? What type of coordinate transformation are you assuming in order to show that the tensor is isotropic?

Do you know the general method for transforming tensor components from one system of coordinates to another?

 

[Raj90]

Hey TSny!

No I am completely new to the subject..that is why I need some help and guidance as to what I should refer to solve the given problem.

 

[TSny]

You need to have some basic knowledge about tensors to solve the problem. We are here to help you once you have made an attempt and you show us your work so far. I will just say that for this problem you essentially need the following:

1. Know the definition of "isotropic tensor". This will tell you what type of coordinate transformation you need to consider.

2. Know the general method of transforming tensor components and apply this method to the specific type of coordinate transformation you are dealing with in this problem.

 

[paranormal]

As a scientist, it is important to understand the properties and applications of Cartesian tensors. In this case, we are looking at the Kronecker delta, which is a fundamental tensor in Cartesian coordinate systems. The Kronecker delta is defined as δij=δji, which means that it takes on the value of 1 when i=j and 0 when i≠j. This tensor has many important properties, one of which is its isotropy.

Isotropy is the property of a tensor that remains unchanged under coordinate transformations. In other words, if we change the coordinate system, the tensor should still have the same values. In the case of the Kronecker delta, we can see that it is isotropic by considering the definition. Since the Kronecker delta only takes on the value of 1 or 0, it will not change under any coordinate transformation. This is because the values of 1 and 0 are independent of the coordinate system being used.

To further prove the isotropy of the Kronecker delta, we can also consider its transformation properties. Under a coordinate transformation, the Kronecker delta can be written as δ'ij=δikδjl, where δik and δjl are the components of the Kronecker delta in the new coordinate system. Since the Kronecker delta is isotropic, its components will also be isotropic, meaning that they will not change under coordinate transformations. Therefore, the Kronecker delta remains unchanged and maintains its isotropy.

In conclusion, the Kronecker delta is an isotropic tensor, meaning that it remains unchanged under coordinate transformations. This property makes it a very useful tensor in many scientific applications, particularly in mechanics and physics. By understanding the properties and proofs of Cartesian tensors like the Kronecker delta, we can better utilize them in our research and experiments.