What is the Transformation Rule for the Moment of Inertia Tensor?

AI Thread Summary
The moment of inertia is indeed a tensor, represented as a 3x3 matrix. Its transformation rule aligns with standard tensor transformation principles, despite the coordinates being inside the integrals defining it. The confusion arises from the integration process, but the inertia tensor still adheres to covariant and contravariant transformation rules. Understanding the transformation involves recognizing that the tensor itself is independent of the specific coordinate system used in the integrals. Clarifying this relationship helps in grasping how the moment of inertia tensor behaves under coordinate transformations.
JTC
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(Forgive me if this is in the wrong spot)

I understand how tensors transform. I can easily type a rule with the differentials of coordinates, say for strain.

I also know that the moment of inertia is a tensor.

But I cannot see how it transforms as does the standard rules of covariant, contravariant, etc.

Because the coordinates are INSIDE the integrals that define the moment of inertia.

yes, I expect it to be a tensor but I cannot see it . Could someone explain?
 
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Do not worry about the integrands. The inertia tensor is simply a 3x3 matrix and it transforms accordingly.
 

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