The analysis reveals that the two seemingly simple symmetry relations, antisymmetry in the first two indices and symmetry in the last two, are actually a highly restrictive combination. The core of the proof is a series of substitutions that lead to the inescapable conclusion that the tensor must be equal to its own negative ( $T_{a b c}=-T_{a b c}$ ). This logical contradiction forces every component of the tensor to be zero, demonstrating that the only mathematical object that can satisfy these constraints is the trivial zero tensor. This problem serves as a powerful illustration of how the properties and form of a tensor can be completely dictated by its symmetry, leading to a unique and often profound result.

<aside> 🧄

✍️Mathematical Proof

$\complement\cdots$Counselor

</aside>

• Contradiction Through Substitution: The proof's core strategy is to use repeated substitutions based on the two given symmetry relations. By applying them in different sequences, we can manipulate the expression for the tensor's components.
• The Power of Mixed Symmetries: The key insight is that the combination of antisymmetry in the first two indices ( $T_{a b c}=-T_{b a c}$ ) and symmetry in the last two ( $T_{a b c}=T_{a c b}$ ) creates a powerful constraint.
• Logical Culmination: The logical conclusion of these substitutions leads to a single, unavoidable result: $T_{a b c}$ must be equal to its own negative.
• The Trivial Solution: The only mathematical value that can be equal to its own negative is zero. This forces every single component of the tensor to be zero, meaning the tensor itself is the zero tensor.
• Insight into Tensors: This problem serves as a great demonstration of how specific symmetry properties can completely define a tensor's form, in this case, forcing it to be the most trivial possible tensor.

✍️Mathematical Proof

‣

<aside> 🧄

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

🧄Proof and Derivation-1

</aside>