True Meaning of 2nd Derivative Test | Symmetric Tensors, Spectral Theorem, Principal Axes Theorem

In this video, we aim to explore the full geometric meaning of the second derivative test through the lens of linear algebra. We'll uncover the power of the spectral theorem and the principal axes theorem as they apply not just to the Hessian matrix, but also to the Cauchy stress tensor, the moment of inertia tensor, the covariance tensor, and any symmetric rank 2 tensor.


Link to the Desmos demo: https://www.desmos.com/3d/9zi8vws9lp

Chapters:
00:00 Intro
00:50 Geometry of 2nd Partial Test
09:39 2nd Partial Test in 3D and Above
12:37 Avoiding 2nd Partial Test
13:26 Symmetric 2-Tensors
16:01 Tensors Transform Like Tensors
22:16 Spectral Theorem/Principal Axes Theorem
23:38 Outro

Music🎵:
Journey of Wind(Taillteann Theme) - Mabinogi
No.3 Morning Folk Song – Esther Abrami    • No.3 Morning Folk Song  
Machina del Diablo - Stephen Rippy
Creeping Spiders - Nat Keefe & BeatMower    • Creeping Spiders  
Driftveil City Funk Cover - Sean R. Hanson    • Pokemon BW - Driftveil City - Funk Cover