Mwaka wa kwanza wa uzamiliGeometry & TopologyImepangwa
Smooth manifolds
Smooth manifolds, tangent and cotangent bundles, vector fields and flows, differential forms, de Rham cohomology, Stokes's theorem and an introduction to Riemannian metrics.
math.geom.smooth-manifolds
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 41 (hadi chini kabisa)
Utajifunza
- Show that spheres, projective spaces and matrix groups are smooth manifolds, using the regular value theorem.
- Compute with tangent vectors, vector fields, differential forms and pullbacks.
- Prove Stokes's theorem for manifolds with boundary and compute the de Rham cohomology of simple manifolds.
Fasili
- A smooth n-manifold is a second-countable Hausdorff space with a maximal atlas of charts to ℝⁿ whose transition maps are smooth.
- The tangent space TₚM can be defined as the space of derivations at p.
- A differential k-form is a smooth section of the k-th exterior power of the cotangent bundle.
Nadharia
- Regular value theorem: the preimage of a regular value of a smooth map is an embedded submanifold.
- Stokes: ∫M dω = ∫{∂M} ω for a compactly supported (n − 1)-form ω on an oriented n-manifold M with boundary.
- Whitney: every smooth n-manifold embeds in ℝ²ⁿ.
- de Rham: de Rham cohomology is isomorphic to singular cohomology with real coefficients.
Mawazo potofu ya kawaida
- Thinking every manifold comes with a preferred embedding in some ℝᴺ.
- Confusing vectors and covectors when changing coordinates.
Ugunduzi wa hiari
- Functors The tangent construction is a functor from smooth manifolds to vector bundles, and the chain rule is precisely the statement that it preserves composition.
- ← Classical mechanics
- ← Differential geometry of curves and surfaces
Njia za kujifunza
Usomaji zaidi
- John M. Lee, Introduction to Smooth Manifolds