Tangent bundles

What is a "tangent"? In school it meant a line touching a curve. In multivariable calculus it became a plane touching a surface. I thought I knew what was going on. Tangent bundles forced me to admit that I did not. What Schuller slowly makes clear is that tangency is not a geometric convenience but a structural necessity once you try to talk about change in a serious way.

I wondered if I had the right mathematical background to grasp this. As it turns out, you need some background, but less than I initially feared. You need to be comfortable with sets and functions, not just as notation but as objects. You need to understand what a smooth manifold is at the level of charts and coordinate changes, not memorized definitions but the idea that local descriptions must agree globally. Multivariable calculus matters, but mainly for partial derivatives and the chain rule, because everything rests on how quantities transform. Linear algebra is essential, especially the idea of a vector space as something defined by operations rather than pictures. I had linear algebra, multivariable calculus and I watched the lectures on manifolds and charts. I decided to dive in.

The problem began the moment I left flat space. On the real line, velocity is just a number. In the plane, it is an arrow you can slide around without thinking too much. But on a curved space, this intuition breaks. You cannot meaningfully compare arrows at different points without extra structure. Even saying that something moves requires care. Move relative to what. Change in which direction. This is where tangent spaces enter, and the tangent bundle is what you get when you refuse to treat them as isolated local hacks.

At each point of a smooth space, you can define a tangent space. This space captures all possible directions in which you could move through that point. Crucially, this definition does not depend on embedding the space in something bigger. Schuller’s insistence here matters. Tangent vectors are not arrows floating in ambient space. They are operators. They act on functions and tell you how those functions change at a point. Direction becomes synonymous with differentiation. This shift is subtle, and once it clicks, it is irreversible.

Now take all these tangent spaces, one for each point, and glue them together in a systematic way. The result is the tangent bundle. It is a space in its own right, larger than the original one, where each point carries not only a position but also a direction. A single element of the tangent bundle answers two questions at once. Where am I, and how am I moving.

What surprised me most is that the tangent bundle is not just bookkeeping. Many physical objects live there naturally. Velocity fields are not functions on space but sections of the tangent bundle. Dynamics does not happen on spacetime alone but on this enlarged space of positions and directions. Once you see this, classical mechanics stops being about particles tracing curves and starts being about flows on tangent bundles.

What you do not need is physical intuition about forces or motion. That comes later. The tangent bundle is built before physics enters the story. This is one of Schuller’s quiet provocations. If you do not know what kind of mathematical object velocity is, you have no business writing down equations of motion.

I find this deeply comforting.