Analysis II

MAT245N1: Analysis II
2026 – First Semester
Schedule: Monday and Wednesday – 14:55–16:35
Room: 1012 - MAT
Course start: 03/09/2026

Syllabus: Real-valued functions of several variables. Differentiability. Integration.

Program:

1. Differentiable functions of several variables. Chain rule. Mean value inequality.

2. Implicit Function Theorem.

3. Higher-order derivatives. Classification of non-degenerate critical points.

4. Lagrange multipliers.

5. Regular curves and parametrizations.

6. Differential forms of degree one.

7. Line integrals. Green’s Theorem. Characterization of exact forms.

8. Double and triple integrals. Change of variables in double and triple integrals.

9. Surface integrals. Stokes’ and divergence theorems for hypersurfaces in ℝⁿ.

Assessment System:
The semester will include three exams, according to the schedule below:

• Exam 1 (P1): 04/27/26 (Monday) – 30 points

• Exam 2 (P2): 05/20/26 (Wednesday) – 30 points

• Exam 3 (P3): 06/24/26 (Wednesday) – 30 points

• Exercise Lists (L) – 10 points

• Special Exam (EE): 06/29/26 (Monday)
  
  

The Substitute Exam is intended exclusively to replace an exam that was not taken, upon presentation of a medical justification or equivalent.
The content of the Substitute Exam will be the same as that of the replaced exam (P1, P2, or P3), corresponding to the syllabus evaluated in that exam.
The exam will be administered in the first class in which the student is able to attend, upon presentation of the justification.

Evaluation Criteria:
Direct approval:
Students with N=P1+P2+P3+L≥60 will be approved with final grade N.

Grades:
Approved: Total grade (N) ≥ 60.
Recovery: Grade between 40 and 59 → eligible for the special exam (EE).

Main References:

1. BARTLE, R. G.: Elements of Real Analysis, Campus Publishing.

2. LANG, S.: Analysis 1. Addison-Wesley.

3. LIMA, E. L.: Curso de Análise, Vol. 2. IMPA.

4. LIMA, E. L.: Análise Real, Vol. 2. SBM.

5. LIMA, E. L.: Análise no ℝⁿ. SBM.

6. MARSDEN, J. E. and TROMBA, A.: Vector Calculus. W. H. Freeman.

7. SPIVAK, M.: Calculus on Manifolds. Addison-Wesley.