Manifolds, Tensor Analysis, and Applications

algebra Banach space bundle map called chart closed compact compute continuous converges coordinates countable curl D₁ defined definition denote dense Df(x diffeomorphism differential e₁ e₂ equations equivalent example exists F₁ F₂ fiber finite dimensional flow fluid formula function given Hamiltonian Hausdorff Hence Hilbert space implies induced inner product integral curve isomorphism lemma Let f Lie derivative linear map M₁ manifold Marsden matrix measure zero metric space n-manifold neighborhood norm one-form open set operator oriented P₁ partitions of unity Proof Proposition prove Riemannian self-adjoint smooth submanifold subset subspace surjective symmetric symplectic tangent tensor theorem topological space topology U₁ U₂ unique v₁ vector bundle vector bundle map vector field vector space w₁ X₁ ду дх