A mixed tensor transforms by applying the Jacobian to its contravariant components and the inverse Jacobian to its covariant components, demonstrating that each index transforms independently based on its type.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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• 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

🧄Proof and Derivation-1

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