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.

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✍️Mathematical Proof

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The Nature of Tensors

• Coordinate-Invariant Properties: Tensors are mathematical objects whose properties, like their physical meaning or behavior, do not depend on the specific coordinate system used to describe them. An operation on a tensor is considered "valid" in tensor analysis if the result of that operation is also a tensor.

The Proof of Invariance

• Contraction as a Tensor Operation: The proof demonstrates that contraction is a valid tensor operation. This is shown by proving that the contracted components ( $S^{\prime}$ ) transform according to the correct tensor transformation rule for a lower-rank tensor, which is the defining characteristic of a tensor.
• Role of the Kronecker Delta: The core of the proof relies on a key identity involving the chain rule. The product of the partial derivative terms for the contracted indices, $\sum_{c^{\prime}}\left(\frac{\partial x^{\prime c^{\prime}}}{\partial x^c}\right)\left(\frac{\partial x^d}{\partial x^{\prime c^{\prime}}}\right)$, simplifies to the Kronecker delta $\left(\delta_c^d\right)$. This delta effectively forces the contracted indices to be the same in the unprimed system, ensuring the contraction is performed correctly regardless of the coordinate system.

The Result

• Contraction Yields a New Tensor: Contraction takes a rank (m, n) tensor and produces a new object of rank ( m-1, n-1 ).
• A Property, Not a Calculation: The final result confirms that contraction is a geometric property of the tensor, not just a numerical procedure. For example, the trace of a matrix (a type ( 1,1 ) tensor) is an invariant scalar value that remains the same regardless of how you rotate or stretch the coordinate axes.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient

🧄Proof and Derivation-1

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