Tensor contraction is a fundamental, coordinate-independent operation. The proof demonstrates this by showing that if you perform a contraction on a tensor and then transform the result into a new coordinate system, the outcome is identical to first transforming the original tensor and then performing the contraction. This is mathematically validated by the chain rule, which simplifies a complex product of partial derivatives into the Kronecker delta. This crucial step ensures that the contraction always yields the same scalar value, regardless of the chosen basis, confirming that the operation is a genuine geometric property of the tensor, not just a calculation tied to a specific coordinate system.
A derivative illustration based on our specific text and creative direction
A derivative illustration based on our specific text and creative direction
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