The weight ($W$), determined by the determinant factor $\left(J^{-1}\right)^W$, dictates their algebraic properties. Addition of two tensor densities is only possible if they share the same type and the same weight, as the determinant factor must be identical to be factored out. Multiplication (outer product) results in a new tensor density whose weight is the sum of the individual weights ( $W_1+W_2$ ), a direct consequence of multiplying the two determinant factors. Finally, contraction (summing over a contravariant and a covariant index) uniquely preserves the weight of the original tensor density, because the transformation factors for the contracted indices cancel each other out, leaving the determinant factor $\left(J^{-1}\right)^W$ unchanged.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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1. Addition : Preserve Type and Weight

   • Condition for Addition: Tensor densities can only be added if they have the same contravariant and covariant ranks (same type, e.g., ($p, q$) ) and the same weight ( $W$ ).
   • Result: The sum is a new tensor density of the exact same type and weight.
   • Mechanism: The transformation factors (both the $\frac{\partial x}{\partial x}$ derivatives and the determinant factor $\left(J^{-1}\right)^W$ ) are identical for both operands, allowing the addition to be factored out, which preserves the overall transformation structure.

2. Multiplication : Weights Are Additive

   • Operation: The product (outer product) of two tensor densities is always well-defined.
   • Type: The rank of the resulting tensor density is the sum of the ranks of the factors, e.g., $\left(p_1+p_2, q_1+q_2\right)$.
   • Weight: The weight of the resulting tensor density is the sum of the individual weights ( $W=W_1+W_2$).
   • Mechanism: The determinant factors multiply: $\left(J^{-1}\right)^{W_1} \cdot\left(J^{-1}\right)^{W_2}=\left(J^{-1}\right)^{W_1+W_2}$. The weight $W$ acts like an exponent in the transformation law.

3. Contraction $C (\cdot)$ : Weight Is Unchanged

   • Operation: Contraction involves setting one contravariant index equal to one covariant index and summing (e.g., $T_{\ldots j \ldots} \rightarrow C_{\ldots}=T_{\ldots j} \ldots...$
   • Type: The rank is reduced by one for both contravariant and covariant indices (e.g., $(p, q) \rightarrow(p-1, q-1)$).
   • Weight: The contraction operation does not change the weight of the tensor density ( $W_{\text {new }}=W_{\text {original }}$).
   • Mechanism: The transformation factors for the contracted indices cancel each other out and become a Kronecker delta ( $\delta_{i_k}^{j j_k}$ ) in the original coordinates:

   $$ \frac{\partial \bar{x}^{\alpha_k}}{\partial x^{i_k}} \frac{\partial x^{j_l}}{\partial \bar{x}^{\beta_l}} \delta_{\alpha_k}^{\beta_l}=\delta_{i_k}^{j_l} $$

This cancellation means the determinant factor $\left(J^{-1}\right)^W$ remains untouched, preserving the weight.

✍️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
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates
22. Finding Arc Length and Curve Length in Spherical Coordinates
23. Solving for Metric Tensors and Christoffel Symbols
24. Metric Tensor and Line Element in Non-Orthogonal Coordinates
25. Tensor vs. Non-Tensor Transformation of Derivatives
26. Verification of Covariant Derivative Identities
27. Divergence in Spherical Coordinates Derivation and Verification
28. Laplace Operator Derivation and Verification in Cylindrical Coordinates
29. Divergence of Tangent Basis Vectors in Curvilinear Coordinates
30. Derivation of the Laplacian Operator in General Curvilinear Coordinates
31. Verification of Tensor Density Operations
32. Verification of the Product Rule for Jacobian Determinants and Tensor Density Transformation
33. ‣

🧄Proof and Derivation-1

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