Lesson plan
Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x , denoted \mathrm d\varphi_x , is, in some sense, the best linear approximation of \varphi near x . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x) , \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N . Hence it can be used to push tangent vectors on M forward to tangent vectors on N . The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi .
Article summary
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x , denoted \mathrm d\varphi_x , is, in some sense, the best linear approximation of \varphi near x . It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x) , \mathrm d\varphi_x\colon T_xM \to T_{\varphi(x)}N . Hence it can be used to push tangent vectors on M forward to tangent vectors on N . The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi .
Key claims
No claims recorded.
Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
No sources recorded.
Key claims and bibliography are generated from the same tracked claims and sources as the article page.