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

The core focus of the source is the application of the $\varepsilon-\delta$-relation to transform the tensor expression $\varepsilon_{i j k} \varepsilon_{j k \ell}$ into a representation using the Kronecker delta. To master this, one must approach complex tensor identities by breaking them down into manageable, step-by-step calculations rather than attempting to solve them all at once. Furthermore, visualising the summation of individual products is essential, as it highlights the specific non-zero terms required to arrive at the final simplified value.

🎬Narrated Video

Description

This illustration provides a visual step-by-step breakdown of how to simplify a tensor contraction involving the Levi-Civita symbol ($\epsilon$) and the Kronecker delta ($\delta$).

The infographic is organized into three primary phases:

1. The Initial Problem

• Goal: The objective is to express the specific quantity $\epsilon_{ijk}\epsilon_{jkl}$ in terms of the Kronecker delta.

• Visual Representation: A complex, interlocking knot labeled with indices $i, j, k, l$ represents the initial state of the tensor interaction.

2. The Method: Visualizing the Summation

This central section details the mathematical process by breaking down the complex identity into a sequence of operations:

• Summation over $j, k$: The process begins by summing over the shared indices.

• Identify Non-Zero Terms: The workflow filters the summation to focus only on terms that do not evaluate to zero.

• Apply Delta Identity: Multiple steps show the application of the delta identity to transform the Levi-Civita symbols into Kronecker deltas ($\delta_{il}$ and $\delta_{ik}$).

• Simplify Result: The summation is condensed, leading toward the final constant.

3. The Result: Non-Zero Terms Define the Value

• Final Output: The complex interaction is simplified to its final form: $\sum_{i=l}^n = 2\delta_{il}$.

• Key Takeaway: The summation of non-zero terms is the only requirement to produce the final value.

• Simplified Visual: The chaotic knot from the beginning is replaced by a cleaner, more symmetrical geometric symbol, representing the "direct mapping signal" rather than "loop-based summation noise".

Comparison of Result by Dimension

Based on the accompanying theoretical mind map, the constant produced by this simplification scales according to the dimension ($n$) following the formula $(n-1)!$.

🧵Related Derivation

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

⚒️Compound Page

Simplify Tensor Products with Epsilon-Delta | IllustraDemoviadean on Notion