Jet (mathematics)

A jet in mathematics is a way to understand how a function changes. Think of a smooth curve. A jet helps us know not just the height at one point, but also how steep it is and how it curves. It makes a special kind of math shape called a Taylor polynomial. This polynomial matches the function perfectly at that point. It helps us study functions by looking closely at their behavior.

Jets are useful because they let mathematicians compare and work with functions more easily. They turn tricky, curvy functions into simpler shapes that are easier to handle. This idea works for functions with many variables and even for curved spaces called manifolds.

Learning about jets connects to many parts of math. It links to differential geometry, which studies shapes and curves. It also connects to the theory of differential equations. These equations describe how things change. By understanding jets, scientists and mathematicians can solve problems about how shapes bend and twist, and how systems change over time.

Jets of functions between Euclidean spaces

Jets help us understand functions by creating simpler versions of them — called polynomials — at each point. Imagine you have a function, like one that tells you the height of a hill at different spots. A jet creates a smooth, simple curve that matches the hill's shape very closely at one specific spot. This makes complex functions easier to study.

When we look at functions between different spaces — for example, from a flat surface to another flat surface — jets still work the same way. They create polynomials that copy the function’s behavior near a chosen point. These jets can be combined in special ways, like multiplying them or "composing" them. This helps mathematicians understand how functions interact with each other.

Jets at a point in Euclidean space: rigorous definitions

A jet in mathematics helps us understand functions by looking at what they do near a specific point. Think of a smooth curve and wanting to describe it closely around one spot using a simple polynomial. Jets do this—they record the function's value and its changes at that point, making a polynomial that matches the function’s behavior there.

Jets are useful because they let us study functions by looking only at what happens close to a point. Whether the functions are simple or complicated, jets help us compare them based on how they match at that point, including how they change. This idea works for many variables and different kinds of functions, making jets a valuable tool in advanced mathematics.

Jets of functions between two manifolds

When we have two smooth manifolds, M and N, we can define the jet of a function f that maps from M to N. To do this, we use local coordinates, but this approach has a drawback: it doesn’t treat jets in a way that is independent of the coordinate system we choose. Because of this, jets of functions between two manifolds are actually part of something called a jet bundle.

Jets of functions from the real line to a manifold

Imagine we have a smooth manifold M with a specific point p in it. We want to look at smooth curves — functions from the real numbers to M — that pass through p. We can group these curves together based on how they behave up to a certain order k at the point p. This grouping is called an equivalence relation.

The k-jet of a curve is just the group or class of curves that behave the same way up to order k at p. The collection of all such k-jets at p forms a space called the k-th-order jet space. As we look at different points in M, these jet spaces fit together to form a bundle over M, known as the k-th-order tangent bundle.

Jets of functions from a manifold to a manifold

For functions that map from one manifold M to another manifold N, we define jets similarly. We consider smooth functions near a point p in M and group them based on how they behave when composed with curves through p. The jet space here is the set of these equivalence classes.

Multijets

A concept introduced by John Mather, multijets are lists of jets taken at different base points. Mather used the multijet transversality theorem in his research on stable mappings.

Jets of sections

In mathematics, a jet of a section is a way to study smooth functions between spaces. Think of a smooth surface and a function that moves along it. The jet of this function shows how the function changes near a specific point, like a simple version of the function around that spot.

The space of these jets at a point is a vector space. This means it follows special rules of addition and multiplication by numbers. As we look at different points on the surface, these spaces line up to form another bundle, called the jet bundle. This helps mathematicians understand more complex structures and changes in functions on surfaces.