3.1 Linearization PROOF | Nonlinear Dynamics

Nonlinear Dynamics mini-series Part 1: https://youtu.be/ySfs8YVMY7Q Part 2: https://youtu.be/I9UEBRya4X0 Part 3: https://youtu.be/6mLCFyEv3Z0 This video shows a formal proof behind linearization for 2D flows: dx/dt = f(x,y) dy/dt = g(x,y) Step 1: Find fixed points. This involves solving for where dx/dt and dy/dt both are equal to 0. Step 2: Approximate f(x,y) and g(x,y) as linear functions at these fixed points. This results in the matrix equation: d/dt X = [A] X. The matrix [A] will contain partial derivatives of f and g. The matrix [A] is also known as the jacobian matrix. Chapters: 0:00 Intro 0:24 Fixed Points 0:44 Multivariable Taylor Series 2:06 Local Coordinate System 4:04 Limitations of Linearization 5:08 Outro Music: Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u -- Patreon: https://www.patreon.com/VirtuallyPassed Instagram: https://www.instagram.com/virtuallypassed/