https://www.chenk.top/en/tags/lie-bracket/Recent content in Lie-Bracket on Chen Kai BlogHugoenSat, 13 Nov 2021 09:00:00 +0000https://www.chenk.top/en/differential-geometry/07-vector-fields-flows/Sat, 13 Nov 2021 09:00:00 +0000https://www.chenk.top/en/differential-geometry/07-vector-fields-flows/<p>A tangent vector lives at one point. It tells you &ldquo;this direction, this speed, right here, right now.&rdquo; It is fundamentally local — pluck it off the manifold and it remembers nothing about its neighbors. A <strong>vector field</strong>, by contrast, is what you get when you let one tangent vector at every point conspire smoothly. It is a velocity prescription on the entire manifold: stand anywhere, and the field tells you where to go. Follow the prescription, and you trace out an <strong>integral curve</strong>. Follow it from every starting point at once, and you get a <strong>flow</strong> — a one-parameter family of diffeomorphisms that drags the whole manifold along itself like a slow river.</p>