Mathematical Analysis II

Recommended Prerequisites

Sequences; Differential and integral calculus of real-valued functions of one real variable (Mathematical Analysis I).

Teaching Methods

Detailed lectures (using occasionally some audio-visual devices) introducing and explaining concepts, principles and theories. Theoretical-practical classes, in which the students with the guidance of the teacher, solve exercises.

Learning Outcomes

The aim of this curricular unit is to o provide the main concepts and techniques of sequences and numerical, power and Fourier series, vectors and the space geometry, and partial derivates, which will be intensively used in the remaining curricular unites.

Work Placement(s)

No

Syllabus

1. Series
1.1 Sequences
1.2 Numerical series
1.3 The integral test and estimates of sums
1.4 The comparison tests
1.5 Alternating series
1.6 Absolute convergence and the ratio and root tests
1.7 Strategy for testing series
1.8 Power series
1.9 Representations of functions as power series
1.10 Taylor and MacLaurin series
1.11 Binomial series
1.12 Taylor formula
1.13 Fourier series in R
1.14 Fourier series of functions defined in a limited interval
1.15 Cosine and sine series
2 Vectors and the Geometry of Space
2.1 Three-dimensional coordinate systems
2.2 Vectors
2.3 The inner product
2.4 The cross product
2.5 Equations of lines and planes
2.6 Cylinders and quadric surfaces
2.7 Cylindrical and spherical coordinates
3. Parcial Derivatives
3.1 Graphs and level curves
3.2 Limits and continuity
3.3 Parcial derivatives
3.4 Tangent planes and linear approximations
3.5 The chain rule
3.6 Directional derivatives and the gradient vector
3.7 Maximum and minimum values
3.8 Lagrange multipliers

Head Lecturer(s)

Amílcar José Pinto Lopes Branquinho

Assessment Methods

Final assessment
Exam: 100.0%

Continuous assessment
2 frequencies: 100.0%

Bibliography

James Stewart  - Cálculo Vol. II -  5ª Edição, São Paulo, Pioneira Thomson Learning, 2001.