Infinitesimal Analysis IV

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Real Analysis II, III; Linear Algebra and Analytic Geometry I, II.

Teaching Methods

Active participation of the students in the theoretical and tutorial classes. This may include presentation in class of part of the homework assigments.

Students' work is closely observed and frequently evaluated by the instructor. The student has access to the results of each evaluation and is encouraged to meet individually with the instructor to discuss his/her performance.

Learning Outcomes

A first contact with Integral Calculus of functions defined in Rn, culminating in the establishment of the four main Throrems of Integral Calculus: Central theorem of curvilinear integral,Riemann-Green Theorem, Stokes theorem and Gauss Theorem. The skills to be developed include learning  the theoretical foundations of Integral Calculus in Rn; capacity of generalization and abstraction; capacity to pose problems and to translate them in mathematical language; calculation capacity; development of oral and written expression.

Work Placement(s)



1. Elements of Jordan measure in Rn.

2. Double integral. Concept and properties. Fubini’s Theorem. Mean Value Theorem. Areas and Volumes. Surface area. Double integral in polar coordinates.

 3. Triple integral. Concept and properties. Formulas of calculus. Triple integral in cylindrical and spherical coordinates. Applications.The concept of integral in Rn.

4. Curvilinear integral of a vector function. Concept and properties. Formulas of calculus. The concept of work. Curvilinear integral of a scalar function. Conservative vector fields. Independence of path. Riemann-Green Theorem. Necessary and sufficient conditions for a field to be conservative.

Generalizations of Riemann-Green Theorem.

5. Change of variable in the double integral.

6. Surface integral. Stokes Theorem.Flux.

7. Gauss Theorem.

8. Gauss Theorem and conservation laws.

Head Lecturer(s)

Maria Paula Martins Serra de Oliveira

Assessment Methods

Approval in this course unit requires to score at least 10 (out of 20). The students that perform, along the semester, the mid-term exams, the test, and/or the homework may be exempted from final examination. The sum of percentages corresponding to these three components is 100%. The other students are evaluated in a final exam which is worth 100%.: 100.0%


J. E. Marsden, Elementary Classical Analysis. Freeman, NY, 1974.


J. E. Marsden, Calculus III. 2nd edition. Springer, NY, 1991.


M. P.  Serra Oliveira, C. Oliveira, Análise Infinitesimal IV, Notas de Curso, 2008