Question on tangent space and jet spaces

In summary, jet spaces are a generalization of tangent spaces, defined as the set of all possible derivatives of a map up to a certain order. Different authors may define jet spaces differently, with some including the point of evaluation in the definition and others not. This can lead to confusion, but it is important to understand the difference between the jet space (set of all jets) and the jet bundle (fibred manifold formed by the jets) when interpreting the definitions.
  • #1
mnb96
715
5
Hello,

I am reading some material related to jet spaces, which at first glance seem to be a generalization of the concept of tangent space.
I am confused about what is the correct definition of a jet space. In particular, given a map ##f: X \rightarrow Y## between two manifolds, what is the k-jet of f at x?

It seems to me that some authors sometimes define it as: $$J^k_xf := \left [ \partial_\alpha f \right(x) ]$$ where ##|\alpha|\leq k## and the multi-indices ##\alpha## contains the indices of the variables with respect to which differentiate. But other authors define it as: $$J^k_xf := \left [x,\, \partial_\alpha f \right(x) ]$$
In other words, they sometimes arbitrarily "attach" also the variable x to the same quantity in the first definition. Which of the two is the correct one?

For example, is the 1-jet space of ##f:\mathbb{R}^m\rightarrow \mathbb{R}^n## at x given by $$J^1_xf=\left[f(x), f_1(x),\ldots,f_m(x)\right]$$ or $$J^1_xf=\left[x,\,f(x), f_1(x),\ldots,f_m(x)\right]$$?
 
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  • #2
I've only dipped my toe into Olver's books on symmetry groups and differential equations so am not yet clear on jet spaces but they are not, as I understand it, fiber bundles. One first constructs a Jet space as a manifold rather than a fiber bundle then determines a specific fibration. So for a functional relation [itex]y=f(x)[/itex] which we may identify with a submanifold of the product space [itex]X\times Y[/itex] the jet spaces are constructed by extending this embedding with variables which, when we later impose constraints, will be identified with the derivatives.
$$ X\times Y \to X\times Y \times Y'\times Y''...$$
The (typically but not necessary functional) relations defining curves in the original product space then by extension define curves in the extended space once one imposes the derivative identifying constraints.

That is, how I understood Peter Olver to define things in Applications of Lie Groups to Differential Equations and others.

And now my understanding gets too fuzzy to continue constructively. There is presumably a fibration of this manifold forming the Jet bundle defined by the matching up the derivatives of curves in the initial [itex]X\times Y[/itex] space to the extended space. This is something I've wanted to sit down for a quiet year and absorb.
 
  • #3
Thanks jambaugh for your help, and especially for pointing me to Olver's work.

I checked Olver's book "Classical invariant theory", and I found there the same confusing "double definition" that seems to propagate in other texts as well. See excerpt below:

olver.JPG
In the first highlighted sentence he clearly defines the jet of a function as $$\left[ f_\alpha(x) \right]$$ (according to my notation), while in the second highlighted sentence he apparently (re)defines the jet of f as $$\left[x, f_\alpha(x) \right]$$

So, are jets defined with that x attached to the prolongation of f or no?
 
  • #4
It depends whether you mean the jet itself or the point in the jet bundle. Similarly as a tangent vector ##v## at a point ##x## or ##(x,v)##.
 
  • #5
I see...so basically I have to interpret the text according to the following analogies:

prolongation (or jet) of f at xtangent vector of f at x
jet space of f at xtangent space of f at x
jet bundle of f tangent bundle of f

This would explain the reason for the "double definition". The first definition (without the attached x) would represent an element of the jet space, while the second one (with the x) represents an element of the jet bundle.

If that's the case, then the confusion arises from using the same term "jet" to denote an element of the jet space or an element of the jet bundle.
 

1. What is a tangent space?

A tangent space is a mathematical concept that represents the set of all possible directions or velocities at a given point on a curved surface or manifold. It is often used in differential geometry to study the behavior of curves and surfaces.

2. How is a tangent space related to a jet space?

A jet space is an extension of the tangent space that includes higher-order derivatives. In other words, while the tangent space only considers the first derivative of a function at a point, the jet space includes all derivatives up to a specified order. This allows for a more comprehensive understanding of the behavior of a function at a specific point.

3. Why are tangent spaces and jet spaces important?

Tangent spaces and jet spaces are important in various fields of mathematics and physics, including differential geometry, calculus of variations, and mechanics. They provide a way to analyze the behavior of functions and surfaces at a specific point, which can be useful in solving problems and making predictions.

4. How do you construct a jet space?

To construct a jet space, one must first choose a point on a manifold and specify the order of the derivatives to be included. Then, using the Taylor series expansion, the jet space can be constructed by considering all possible combinations of the chosen derivatives at that point. This results in a vector space that represents the jet space at that point.

5. Can you give an example of how tangent spaces and jet spaces are used in real-world applications?

One example of how tangent spaces and jet spaces are used in real-world applications is in the study of fluid dynamics. The velocity of a fluid at a specific point can be represented by a vector in the tangent space, and the higher-order derivatives can provide information about the acceleration and curvature of the fluid at that point. This can be useful in predicting the behavior of the fluid in different scenarios, such as in weather forecasting or designing aerodynamic structures.

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