 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: T^{n}, S^{n},
RP^{n}, submanifolds, product manifolds.
 The classification of 1dimensional 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
R^{N}.
 Whitney's theorem for compact manifolds into
R^{2n+1}.
 Whitney's theorem for arbitrary manifolds into
R^{N}.
 Whitney's theorem for arbitrary manifolds into
R^{2n+1}.
 The proof of Sard's theorem.
 Manifolds with boundary.
 The classification of 1manifolds with boundary.
 The boundary of g^{1}(y) when the domain manifold
has boundary.
 S^{n} is not a smooth retract of
D^{n+1}.
 The smooth approximation theorem where the target is
R^{N}.

The normal bundle to an embedding in
R^{N}.

 The tubular neighborhood theorem.
 The smooth approximation theorem where the target is a compact
manifold.
 S^{n} is not a continuous retract of
D^{n+1}.
 The Brower fixed point theorem (and simple applications).
 Smooth maps of a low dimensional manifold into S^{n}
are homotopic to constant maps.
 S^{n} is not contractible.
 S^{n} is not homemomorhic to S^{m}
for n<m.
 R^{n} is not homemomorhic to
R^{m} for n<m.
 Lie groups and some examples.
 S^{3} 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 Leftinvariant 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 A^{p}(V).
 The wedge product and its elementary properties.
 Basis and dimension of A^{p}(V).
 Differential forms on a manifold.
 Uniqueness and existance of the exterior derivative d.
 The case of R^{3}: 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
R^{N}.
 Orientability (several definitions!).
 Integration of compactly supported top forms on arbitrary orientable
manifolds.
 Stokes' theorem.
 The 3dimensional cases of Stokes' theorem.

The planimeter.

Poincare's lemma and its proof.
 The definition of deRham 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.
