The derivation of the tensor transformation properties for mixed tensors establishes the rule that a mixed tensor of type (n, m), having n contravariant (upper) indices and m covariant (lower) indices, transforms according to the independent action of each index type. The core principle is that a scalar formed by contracting the mixed tensor with appropriate covariant and contravariant vectors must be invariant under coordinate changes. This leads to the transformation rule: each contravariant index transforms with a factor of the Jacobian $\left(\frac{\partial y^{a'}}{\partial y^a}\right)$, and each covariant index transforms with a factor of the inverse Jacobian $\left(\frac{\partial y^b}{\partial y^{b'}}\right)$. The overall transformation is the product of n Jacobian factors and m inverse Jacobian factors.

Key takeaways

• Mixed Tensors are Hybrids: A mixed tensor of type ( $n, m$ ) has both contravariant (superscripted) and covariant (subscripted) indices. It's essentially a multi-dimensional quantity that combines both types of transformation behavior.
• Contravariant vs. Covariant Transformations: The key difference lies in the transformation factor.
• Contravariant components transform with the Jacobian matrix, which is the matrix of partial derivatives $\left(\partial y^{a^{\prime}} / \partial y^a\right)$.
• Covariant components transform with the inverse Jacobian matrix, the inverse of that same matrix $\left(\partial y^b / \partial y^{b^{\prime}}\right)$.
• The Invariant Scalar: The derivation for a mixed tensor relies on the principle that a scalar quantity, formed by combining a mixed tensor with appropriate vectors, must remain invariant regardless of the coordinate system. This invariance acts as a constraint that reveals the transformation rule for the tensor itself.
• Independent Index Transformation: The final transformation rule for a mixed tensor is a product of the individual transformation rules for each index. Each contravariant index multiplies by its corresponding Jacobian term, and each covariant index multiplies by its corresponding inverse Jacobian term. This means each index transforms independently based on its type and position.

✍️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. Metric Determinant and Cross Product in Scaled Coordinates
34. Vanishing Divergence of the Levi-Civita Tensor
35. Curl and Vector Cross-Product Identity in General Coordinates
36. Curl of the Dual Basis in Cylindrical and Spherical Coordinates
37. Proof of Covariant Index Anti-Symmetrisation
38. Affine Transformations and the Orthogonality of Cartesian Rotations
39. Fluid Mechanics Integrals for Mass and Motion
40. Volume Elements in Non-Cartesian Coordinates (Jacobian Method)
41. Young's Modulus and Poisson's Ratio in Terms of Bulk and Shear Moduli
42. Tensor Analysis of the Magnetic Stress Tensor
43. Surface Force for Two Equal Charges
44. Total Electromagnetic Force in a Source-Free Static Volume
45. Proof of the Rotational Identity
46. Finding the Generalized Inertia Tensor for the Coupled Mass System
47. Tensor Form of the Centrifugal Force in Rotating Frames
48. Derivation and Calculation of the Gravitational Tidal Tensor
49. Conversion of Total Magnetic Force to a Surface Integral via the Maxwell Stress Tensor
50. Verifying the Inhomogeneous Maxwell's Equations in Spacetime

🧄Proof and Derivation-1

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