Jet (mathematics)

In mathematics, a jet is a special way to understand how a function changes. Imagine you have a smooth curve, and you want to know not just its height at one point, but also how steep it is and how it curves. A jet takes all that information and creates a polynomial that matches the function perfectly at that point. This polynomial is called the Taylor polynomial, and it helps us study functions by looking at their behavior up close.

Jets are important because they let mathematicians compare and work with functions in a more structured way. They turn complicated, curvy functions into simpler polynomial shapes that are easier to handle. This idea can be expanded from simple functions to more complex ones with many variables, and even to curved spaces called manifolds.

The study of jets connects to many areas of math, like differential geometry, which looks at shapes and curves, and the theory of differential equations, which describes how things change. By understanding jets, scientists and mathematicians can solve problems about how shapes bend and twist, and how systems evolve 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 is useful because it 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, which helps mathematicians understand how functions interact with each other.

Jets at a point in Euclidean space: rigorous definitions

The concept of a jet in mathematics helps us understand functions by looking at their behavior near a specific point. Imagine you have a smooth curve, and you want to approximate it near a point using a simple polynomial. Jets do exactly this—they capture the value of the function and its derivatives at a point, creating a polynomial that matches the function’s behavior there.

Jets are useful because they let us study functions by focusing on their local properties. Whether we’re working with simple functions or more complex ones, jets provide a way to compare functions based on how they agree at a point, including their rates of change. This idea can be extended to many variables and different types of functions, making jets a powerful 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. Imagine you have a smooth surface and a function that moves along it. The jet of this function captures how the function changes near a specific point, much like a simplified version of the function around that spot.

The space of these jets at a point forms a vector space, which 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.