🧄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

📢Tensors Define 3D Vector Direction

🧣Example-to-Demo

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

🍁Rotational Dynamics via Tensor Calculus

Description

These sources outline a pedagogical framework for mastering complex rotational mechanics by bridging abstract tensor mathematics with Python-based visualizations. The curriculum progresses from the fundamental properties of the Levi-Civita symbol—acting as a "logic gate" for vector orientations—to the practical calculation of physical quantities like torque and angular momentum using Einstein notation. A core focus is placed on the inertia tensor, illustrating how mass distribution influences rotational resistance and leads to phenomena like vector misalignment and dynamic wobbling. By integrating computational demos, the materials transform theoretical expressions, such as the Inertia Ellipsoid and 3D coupling, into intuitive geometric models that clarify how objects resist or respond to rotation in three-dimensional space.

⚒️Compound Page

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