- The differential for f:V-->W.
- The differential matrix.
- The chain rule.
- The inverse function theorem, the "jelly cube" principle.
- The implicit function theorem.
- Manifolds through atlases.
- Uniqueness of the extension to a maximal atlas.
- Functional structures and morphisms between them.
- Manifolds through functional structures.
- The equivalence of the two definitions of manifolds.
- Smooth maps between manifolds.
- Simple examples: Tn, Sn,
RPn, submanifolds, product manifolds.
- The classification of 1-dimensional manifolds (formulation only).
- Tangent vectors as equivalence classes of paths.
- Tangent vectors as derivations.
- Hadamard's lemma and the equivalence of the two definitions of tangent
vectors.
- The tangent space and its dimension.
- Pushing forward tangent vectors and the differential.
- Definitions and examples of immersions, submanifolds, embeddings,
submersions.
- The local structure of immersions.
- The functional structure induced on an embedded submanifold.
- The local structure of submersions.
- Regular values and Sard's theorem.
- The inverse image of a regular value is an embedded submanifold.
- The orthogonal group O(m,n) as a manifold.
- Transversal submanifolds and the local structure of transverse
intersections;
the "lens spaces" example.
- The "partition of unity" theorem (formulation only).
- The "smooth Tietze theorem".
- The "tangent bundle" TM is a manifold.
- Existence of proper functions.
-
Closedness of proper functions.
- Whitney's theorem for compact manifolds into
RN.
- Whitney's theorem for compact manifolds into
R2n+1.
- Whitney's theorem for arbitrary manifolds into
RN.
- Whitney's theorem for arbitrary manifolds into
R2n+1.
- The proof of Sard's theorem.
- Manifolds with boundary.
- The classification of 1-manifolds with boundary.
- The boundary of g-1(y) when the domain manifold
has boundary.
- Sn is not a smooth retract of
Dn+1.
- The smooth approximation theorem where the target is
RN.
-
The normal bundle to an embedding in
RN.
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- The tubular neighborhood theorem.
- The smooth approximation theorem where the target is a compact
manifold.
- Sn is not a continuous retract of
Dn+1.
- The Brower fixed point theorem (and simple applications).
- Smooth maps of a low dimensional manifold into Sn
are homotopic to constant maps.
- Sn is not contractible.
- Sn is not homemomorhic to Sm
for n<m.
- Rn is not homemomorhic to
Rm for n<m.
- Lie groups and some examples.
- S3 and SU(2) are diffeomorphic.
- Vector fields and smooth derivations.
-
Vector fields and flows.
- The Lie brackect of vector fields.
-
The "flows" interpretation of the Lie bracket.
- The tangent at the identity and Left-invariant vector fields.
- The Lie algebra.
- The Lie algebra of SU(2).
-
The classification of compact connected Abelian Lie
groups.
-
Vector bundles, trivial vector bundles and parallelizable
manifolds.
- The linear space Ap(V).
- The wedge product and its elementary properties.
- Basis and dimension of Ap(V).
- Differential forms on a manifold.
- Uniqueness and existance of the exterior derivative d.
- The case of R3: div, grad, curl, and all
that.
- The pullback of differential forms and its elementary properties.
- The Jacobian as it arises for pullbacks of differential forms.
- Integration of compactly supported top forms on
RN.
- Orientability (several definitions!).
- Integration of compactly supported top forms on arbitrary orientable
manifolds.
- Stokes' theorem.
- The 3-dimensional cases of Stokes' theorem.
-
The planimeter.
-
Poincare's lemma and its proof.
- The definition of de-Rham cohomology and some simple examples (also in
homework assignments!).
- The Hodge star operator (only in the Euclidean case).
- Integration by parts.
- The action principle for electromagnetism.
- The equations dJ=0, dF=0 and d*F=J.
- Maxwell's equations.
-
Links and linking numbers.
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