And now, finally, I resume and hopefully finish what was meant to be a simpler and less stressful A-to-Z for last year. I’m feeling much better about my stress loads now and hope that I can soon enjoy the feeling of having a thing accomplished.

This topic is one of many suggestions that Elkement, one of my longest blog-friendships here, offered. It’s a creation that sent me back to my grad school textbooks, some of those slender paperback volumes with tiny, close-set type that turn out to be far more expensive than you imagine. Though not in this case: my most useful reference here was V I Arnold’s **Ordinary Differential Equations**, stamped inside as costing $18.75. The field is full of surprises. Another wonderful reference was this excellent set of notes prepared by Jodin Morey. They would have done much to help me through that class.

# Tangent Space

Stand in midtown Manhattan, holding a map of midtown Manhattan. You have — not a tangent space, not yet. A tangent plane, representing the curved surface of the Earth with the flat surface of your map, though. But the tangent space is near: see how many blocks you must go, along the streets and the avenues, to get somewhere. Four blocks north, three west. Two blocks south, ten east. And so on. *Those* directions, of where you need to go, are the tangent space around you.

There is the first trick in tangent spaces. We get accustomed, early in learning calculus, to think of tangent lines and then of tangent planes. These are nice, flat approximations to some original curve. But while we’re introduced to the tangent space, and first learn examples of it, as tangent planes, we don’t stay there. There are several ways to define tangent spaces. One recasts tangent spaces in group theory terms, describing them as a ring based on functions that are equal to zero at the tangent point. (To be exact, it’s an ideal, based on a quotient group, based on two sets of such functions.)

That’s a description mathematicians are inclined to like, not only because it’s far harder to imagine than a map of the city is. But this ring definition describes the tangent space in terms of what we can do with it, rather than how to calculate finding it. That tends to appeal to mathematicians. And it offers surprising insights. Cleverer mathematicians than I am notice how this makes tangent spaces very close to Lagrange multipliers. Lagrange multipliers are a technique to find the maximum of a function subject to a constraint from another function. They seem to work by magic, and tangent spaces will echo that.

I’ll step back from the abstraction. There’s relevant observations to make from this map of midtown. The directions “four blocks north, three west” do not represent any part of Manhattan. It describes a way you might move in Manhattan, yes. But you could move in that direction from many places in the city. And you could go four blocks north and three west if you were in any part of any city with a grid of streets. It is a vector space, with elements that are velocities at a tangent point.

The tangent space is less a map showing where things are and more one of how to get to other places, closer to a subway map than a literal one. Still, the topic is steeped in the language of maps. I’ll find it a useful metaphor too. We do not make a map unless we want to know how to find something. So the interesting question is what do we try to find in these tangent spaces?

There are several routes to tangent spaces. The one I’m most familiar with is through dynamical systems. These are typically physics-driven, sometimes biology-driven, problems. They describe things that change in time according to ordinary differential equations. Physics problems particularly are often about things moving in space. Space, in dynamical systems, becomes “phase space”, an abstract universe spanned by all of the possible values of the variables. The variables are, usually, the positions and momentums of the particles (for a physics problem). Sometimes time and energy appear as variables. In biology variables are often things that represent populations. The role the Earth served in my first paragraph is now played by a manifold. The manifold represents whatever constraints are relevant to the problem. That’s likely to be conservation laws or limits on how often arctic hares can breed or such.

The evolution in time of this system, though, is now the tracing out of a path in phase space. An understandable and much-used system is the rigid pendulum. A stick, free to swing around a point. There are two useful coordinates here. There’s the angle the stick makes, relative to the vertical axis, . And there’s how fast the stick is changing, . You can draw these axes; I recommend as the horizontal and as the vertical axis but, you know, you do you.

If you give the pendulum a little tap, it’ll swing back and forth. It rises and moves to the right, then falls while moving to the left, then rises and moves to the left, then falls and moves to the right. In phase space, this traces out an ellipse. It’s your choice whether it’s going clockwise or anticlockwise. If you give the pendulum a huge tap, it’ll keep spinning around and around. It’ll spin a little slower as it gets nearly upright, but it speeds back up again. So in phase space that’s a wobbly line, moving either to the right or the left, depending what direction you hit it.

You can even imagine giving the pendulum *just* the right tap, exactly hard enough that it rises to vertical and balances there, *perfectly* aligned so it doesn’t fall back down. This is a special path, the dividing line between those ellipses and that wavy line. Or setting it vertically there to start with and trusting no truck driving down the street will rattle it loose. That’s a very precise dot, where is exactly zero. These paths, the trajectories, match whatever walking you did in the first paragraph to get to some spot in midtown Manhattan. And now let’s look again at the map, and the tangent space.

Within the tangent space we see what changes would change the system’s behavior. How much of a tap we would need, say, to launch our swinging pendulum into never-ending spinning. Or how much of a tap to stop a spinning pendulum. Every point on a trajectory of a dynamical system has a tangent space. And, for many interesting systems, the tangent space will be separable into two pieces. One of them will be perturbations that don’t go far from the original trajectory. One of them will be perturbations that do wander far from the original.

These regions may have a complicated border, with enclaves and enclaves within enclaves, and so on. This can be where we get (deterministic) chaos from. But what we usually find interesting is whether the perturbation keeps the old behavior intact or destroys it altogether. That is, how we can change where we are going.

That said, in practice, mathematicians don’t use tangent spaces to send pendulums swinging. They tend to come up when one is past studying such petty things as specific problems. They’re more often used in studying the ways that dynamical systems can behave. Tangent spaces themselves often get wrapped up into structures with names like tangent bundles. You’ll see them proving the existence of some properties, describing limit points and limit cycles and invariants and quite a bit of set theory. These can take us surprising places. It’s possible to use a tangent-space approach to prove the fundamental theorem of algebra, that every polynomial has at least one root. This seems to me the long way around to get there. But it is amazing to learn that is a place one can go.

I am so happy to be finally finishing Little 2021 Mathematics A-to-Z. All of this project’s essays should be at this link. And all my glossary essays from every year should be at this link. Thank you for reading.

Wow – thanks for picking up another suggestion of mine again!! Great post!

Maybe tangent spaces and other concepts explained in Differential Geometry classes should be part of undergrad math classes. It’s so hard to unlearn later what you believed a vector was (or a gradient etc.)! ;-)

BTW, I am still working with Frankel’s book :-)

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I’m very glad for the suggestion; it helped me to learn a bit more about why they work and why I was having trouble with them before. I suspect the biggest gap I ever had in my mathematics training was in geometry; I went to such strongly calculation-oriented programs that I was always weak on this side of things.

Glad you’re enjoying Frankel! I keep going back to that for, like, almost every essay here that gives me any excuse.

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