🧄Simplifying Levi-Civita and Kronecker Delta Identities (LC-KDI)

This derivation is that the contraction of two Levi-Civita symbols results in a specific combination of Kronecker deltas, effectively transforming a product of antisymmetric tensors into a symmetric scale. By utilizing the cyclic property ( $\varepsilon_{i j k}=\varepsilon_{j k i}$ ) and the standard identity for a single shared index, we can reduce the expression through successive index summation. In 3-dimensional space, summing over two shared indices-as seen in the expression $\varepsilon_{i j k} \varepsilon_{j k \ell}$-collapses the product into $2 \delta_{i \ell}$. This relationship is a fundamental tool in vector calculus and tensor analysis, often used to simplify complex crossproduct identities (like the BAC-CAB rule) into manageable algebraic terms.

🎬Narrated Video

• Demo

🎬Comparative Analysis of Dimensional Scaling

📢IllustraDemo

• Illustration

📢Simplify Tensor Products with Epsilon-Delta (TP-ED)

🧣Example-to-Demo

• Flowchart and Mindmap

🧣Levi-Civita and Kronecker Delta Identities (LC-KDI)

🍁Dimensional Scaling of Tensor Identities

Description

The documents outline a Python-based computational framework for simplifying tensor contractions, specifically focusing on the relationship between the Levi-Civita symbol ( $\epsilon$ ) and the Kronecker delta ( $\delta$ ) across different dimensions. By moving from 2D to 5D, the system demonstrates that the resulting scalar constants follow the factorial-based general formula $(n-1)!$, where $n$ represents the dimension. This process simplifies complex, highinteraction summations into efficient "direct mapping signals". These mathematical results are then mapped to geometric equivalents: for instance, the 3D identity $2 \delta_{i l}$ corresponds to the BAC-CAB vector identity, while 4D results relate to hyper-volume scaling.

⚒️Compound Page

Simplifying Levi-Civita and Kronecker Delta Identities | Proof and Derivationviadean on Notion