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Calculus on Manifolds
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Although a manifold, such as the sphere, is well defined without considering it as a subspace of a higher-dimensional space, it is often of interest to do so. In particular, when embedding a manifold in a higher-dimensional space it will turn out that some properties, such as the metric tensor, may be inherited from it. Before we can discuss this properly, we need to go through some of the underlying concepts. We start by looking at a map f from a manifold M to another manifold N. For any given point p in M, the map f induces a natural map f*, called the pushforward, between the tangent spaces TpM and Tf(p)N. As a tangent vector is a directional derivative, the pushforward should bring a directional derivative of functions on M to a directional derivative of functions on N. Taking a vector V in TpM, we define the pushforward by its action on a function φ on N as () (f*V)φ=V(φ∘f),
Decomposition of stochastic flow and an averaging principle for slow perturbations
Published in Dynamical Systems, 2020
Diego Sebastian Ledesma, Fabiano Borges da Silva
Let be smooth vector fields on , and let , be continuous semimartingales over a filtered probability space . Consider the solution flow of where the symbol ° means that we have a Stratonovich SDE and as usual the chance element ω is omitted. Let be the solution flow of where the linear map is the pushforward associated with the diffeomorphism , and given by where . In local coordinate , we can write where
Applications of periodic unfolding on manifolds
Published in Applicable Analysis, 2018
Proof By the very definition of pullback and pushforward, we obtain for ,