Your lecture notes will cover all the material
(except for a few topics assigned for self-study) in the syllabus. The chapters, whenever mentioned below,
provide more extensive explanations, and lots of exercises for you to work on.
Norms on vector spaces, metrics, metrics induced by norms
The Cauchy–Schwarz inequality
(Open) balls, interior points, open sets: Section 8.2 of Apostol, Volume II
Sequences and convergence, limit points of a set, limits versus limit points
Review of basic set theory and countable sets, the definition of uncountability
Dense subsets, countable dense subsets of Rn
Description of the open subsets of the real line
Limits of the values of vector-valued functions; CAUTION: Treatment of this is somewhat non-rigorous in Apostol!
Continuity of sums, products, etc., of continuous functions, examples: Sections 8.4 and 8.5 of Apostol, Volume II
The concepts of compactness and uniform continuity
Properties of compact sets
The Heine–Borel theorem
Cauchy sequences and the completeness of Euclidean space
Differentiability in Rn: Sections 8.11 and 8.18 of Apostol, Volume
II
The Chain Rule
The relation between continuity of partial derivaties and differentiability: Sections 8.13
and relevant exercises in Section 8.14 of Apostol, Volume II
Taylor's Theorem
Critical points, points of local extremum, non-degenerate critical points and the Hessian: Sections 9.11–9.13 of Apostol,
Volume II
Step functions: Sections 11.2 and 11.31 of Apostol, Volume II
Multiple integrals: Sections 11.3–11.6, Section 11.31 and relevant exercises from Sections 11.9 and 11.15 of Apostol,
Volume II
The change-of-variables formula, plane polar and spherical polar coordinates
Parametric manifolds and smooth embedded manifolds
Brief survey of the exterior product of finite-dimensional vector spaces
Differential forms on open sets of Rn
The integral of a differential form, and its independence of parametrization
Orientation: The recommended sources, for the detail in which this was covered, are your lecture notes.
Stokes' Theorem for regular parametric manifolds
Special cases of Stokes' Theorem: The theorem in Apostol, Section 12.11; Green's Theorem
The notion of C-differentiability and the Cauchy–Riemann condition: Gamelin, Sections II.2,
II.3
Polynomials, rational functions and power series
The holomorphic sine, cosine and exponential functions: Gamelin, Sections I.5, I.8