Pushforward

Definition

Pushforward is a mathematical operation that takes a measure or current defined on one space and transfers it to another space via a smooth map. This process allows for the analysis of geometric properties and integrations on the target space, providing a way to relate different structures through mapping. Understanding pushforward is crucial in connecting concepts like currents and flat chains, where it plays a vital role in the transformation and manipulation of these objects across different dimensions.

5 Must Know Facts For Your Next Test

1. The pushforward of a current under a smooth map is computed using the pullback of the corresponding forms, showcasing how currents interact with different manifolds.
2. Pushforward can also be applied to flat chains, where it helps in defining how these chains are transformed across various dimensions.
3. The operation respects certain properties, such as linearity, meaning the pushforward of a sum of currents is the sum of their respective pushforwards.
4. In terms of measures, the pushforward creates a new measure on the target space by transforming the original measure through the mapping.
5. Pushforward plays an essential role in integration theory, allowing integrals over one space to be expressed as integrals over another space through change of variables.

Review Questions

• How does the pushforward operation help in understanding the relationship between currents and their geometrical properties?
  • The pushforward operation allows currents defined on one manifold to be analyzed in the context of another manifold through a smooth mapping. By transferring currents, we can explore how geometric properties such as orientation and volume change under this mapping. This understanding is key in studying interactions between different dimensional spaces and analyzing how their properties are preserved or altered.
• In what ways does pushforward relate to flat chains when considering transformations between spaces?
  • Pushforward relates to flat chains by enabling the transformation of these chains when mapped from one space to another. When applying pushforward to flat chains, we can evaluate how oriented simplices interact with changes in dimensions and mappings. This connection helps facilitate computations involving homology and other topological features, ensuring that we can analyze geometric constructs across different settings.
• Evaluate the importance of the pushforward operation in integration theory and its implications for changing variables in multiple dimensions.
  • The pushforward operation is crucial in integration theory as it provides a systematic way to relate integrals over different spaces via smooth mappings. By enabling changes of variables, it allows mathematicians to compute integrals in more convenient spaces without losing information about the original measure. This becomes especially important in higher dimensions, where direct evaluation may be challenging, thus reinforcing the need for techniques like change of variables in multivariable calculus and beyond.

Related terms

Current:

A current is a generalized notion of integration that extends the idea of differential forms, allowing for the representation of oriented measures on a manifold.

Flat Chain: A flat chain is a formal sum of oriented simplices with integer coefficients, which represents a geometric object in a given dimension.

Smooth Map: A smooth map is a function between manifolds that has continuous derivatives of all orders, preserving the differentiable structure during transformations.