🧄Proving the Cross Product Rules with the Levi-Civita Symbol (CPR-LCS)

The transition demonstrates how the Levi-Civita symbol, $\varepsilon_{i j k}$, acts as a compact bookkeeping device for the geometry of three-dimensional space. By expanding the summation over the indices, we see that the cross product of any two basis vectors $e_j$ and $e_k$ is governed by the cyclic symmetry of the indices: a positive unit vector results from a cyclic permutation (e.g., $1 \rightarrow 2 \rightarrow 3$ ), a negative vector from an anti-cyclic one, and a zero result occurs whenever indices are repeated. This proves that the abstract index notation is perfectly consistent with the standard right-hand rule and the fundamental orthogonality of the Cartesian basis.

🎬Narrated Video

🎬From Indices to Inertia-Visualizing Rotation via Tensor Mechanics (IIR-TM)

📢IllustraDemo

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🧣Example-to-Demo

🧣Levi-Civita and Cross Product (LC-CP)

⚒️Compound Page

Proving the Cross Product Rules with the Levi-Civita Symbol | Proof and Derivationviadean on Notion