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Presentation
Presentation
The Calculus II curricular unit introduces the essential concepts and tools of differential and integral calculus of functions from Rn in Rm, generalizing many of the topics on real functions of a real variable covered in the Calculus I curricular unit. We aim to extend the mathematical training started in the first semester through the development of the ability to abstract and logical reasoning in formulating and solving multidimensional problems, formalizing them with rigor and precision, but without neglecting the virtues of intuitive thinking. The main objective is to provide an introduction to multidimensional Mathematical Analysis, showing the rigorous side of Calculus and establishing the theoretical foundations for further studies in Science and Engineering.
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Class from course
Class from course
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Degree | Semesters | ECTS
Degree | Semesters | ECTS
Bachelor | Semestral | 5
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Year | Nature | Language
Year | Nature | Language
1 | Mandatory | Português
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Code
Code
ULHT7037-714
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Prerequisites and corequisites
Prerequisites and corequisites
Not applicable
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Professional Internship
Professional Internship
Não
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Syllabus
Syllabus
1. Space Rn 2. Curves and paths 3. Real functions of several real variables 4. Vector functions of several real variables 5. Line integrals of scalar fields 6. Line integrals of vector fields 7. Double integrals of scalar fields
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Objectives
Objectives
- Know the algebraic, topological and geometric structures of Rn - Parameterize a curve in Rn from its cartesian equation - Know some cartesian equations of conics and quadrics - Understand the concepts and definitions related to functions from Rn to Rm: limits, continuity, partial derivatives, directional derivatives and differentiability - Apply the chain rule in a context of functions with several variables - Compute line integrals of scalar fields and vector fields - Know how to use line integrals in applications to Physics - Compute double integrals using Fubini's theorem - Compute volumes limited by the graph of a continuous function - Determine and classify the local extrema of a real function of several variables - Find an appropriate change of variables and apply the change of variables formula for double integrals - Apply Green's theorem in the resolution of problems involving line in
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Teaching methodologies
Teaching methodologies
Series of exercises will be proposed to consolidate students' knowledge and develop their problem-solving skills. In each class, students are encouraged to work on questions related to the topics covered and to present the results of their individual work in the following session. Throughout the course, particular emphasis is placed on the fundamental idea that solving exercises is not an end in itself, but rather a means of achieving a deeper understanding of the underlying mathematical concepts. Mastery of these concepts enables students to apply the acquired techniques to the solution of more advanced engineering and scientific problems. To further support learning and promote student engagement, short videos on selected course topics, historical notes, mathematical curiosities, recreational problems, and real-world applications will be made available throughout the course.
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References
References
Anton, H., Bivens, I., & Davis, S. (2012). Calculus (10th ed.). John Wiley & Sons. Sarrico, C. (2009). Cálculo Diferencial e Integral para Funções de Várias Variáveis. Esfera do Caos. Stewart, J., Clegg, D., & Watson, S. (2020). Calculus (9th ed.). Cengage.
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Assessment
Assessment
Descrição
Ponderação
Teste 1
40%
Teste 2
50%
TPC/Participação
10%
Global
90% + 10%
Exame final
100%
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Mobility
Mobility
Yes
