Is this quantity a tensor and why?

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Therefore, in order to make T^{jk} a tensor, B^{ijk} can be multiplied by a factor of \frac{1}{\sqrt{g}} and have its indices lowered using the metric tensor g. In summary, T^{jk} is a tensor when B^{ijk} is multiplied by \frac{1}{\sqrt{g}} and its indices are lowered using g.
  • #1
zouzou0
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T[itex]^{jk}[/itex] = [itex]\frac{1}{\sqrt{g}}[/itex][itex]\frac{\partial}{\partial x^{i}}[/itex] ([itex]\sqrt{g}[/itex]B[itex]^{ijk}[/itex])

Given B[itex]^{ijk}[/itex] is a tensor
Find if T[itex]^{jk}[/itex] is a tensor and explain why and if not what can be done to
B[itex]^{ijk}[/itex] to make T[itex]^{jk}[/itex] a tensor ?

I tried to solve this but i think i'am missing some rules !

I think T[itex]^{jk}[/itex] looks like a div of B = B[itex]^{ijk}[/itex],i
i tried to substitute B[itex]^{ijk}[/itex],i with the derivative but i end up with a big quantity with christoffel symbols

any help please!
 
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  • #2
It's the divergence of a rank 3 (0,3) tensor which is a contraction from the covariant derivative of a tensor. Through contraction, the tensor/pseudo-tensor character is preserved, so...
 
  • #3
thank you for the reply.
I found this and i guess it is your answer!

The divergence of a given contravariant tensor
results from the expression of the covariant
derivative of that tensor, and due to the contraction,
the divergence will be a tensor of a rank less by two
units.
 

1. What is a tensor?

A tensor is a mathematical object that describes the relationships between different quantities in a multi-dimensional space. It is represented by an array of numbers and follows specific transformation rules under coordinate changes.

2. How do you determine if a quantity is a tensor?

To determine if a quantity is a tensor, we need to check if it follows the transformation rules of a tensor under coordinate changes. This includes checking for its transformation properties and whether it is covariant, contravariant, or mixed.

3. Why is it important to know if a quantity is a tensor?

Knowing if a quantity is a tensor is important because tensors have special properties that allow us to simplify and generalize mathematical equations. This can make solving complex problems in physics, engineering, and other fields more efficient and accurate.

4. Can a quantity be a tensor in one coordinate system but not in another?

Yes, a quantity can be a tensor in one coordinate system but not in another. This is because tensors follow specific transformation rules that may differ between coordinate systems. A quantity may appear to be a tensor in one coordinate system but not in another due to changes in basis vectors and coordinate axes.

5. How is a tensor different from a scalar or a vector?

A tensor is different from a scalar or a vector in that it can represent relationships between multiple quantities in a multi-dimensional space, while a scalar represents a single value and a vector represents a direction and magnitude in a specific coordinate system. Tensors can also have different transformation properties compared to scalars and vectors.

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