2024
Mathematics II
Name: Mathematics II
Code: MAT12237L
6 ECTS
Duration: 15 weeks/156 hours
Scientific Area:
Mathematics
Teaching languages: Portuguese
Languages of tutoring support: Portuguese
Sustainable Development Goals
Learning Goals
Provide the students with (i) a overview of the most important tools in Linear Algebra and Differential
Calculus in ℝn that will be useful in their forthcoming studies, and (ii) apply the different knowledge into
the resolution of problems.
Calculus in ℝn that will be useful in their forthcoming studies, and (ii) apply the different knowledge into
the resolution of problems.
Contents
I Linear Algebra
1. Vector spaces
2. Linear functions
3. Matrices and Linear Systems of Equations .
4. Determinants Permutations.
5. Eigenvalues and eigenvectors Definitions. The caracteristic polynomial. Algebraic and geometric
multiplicities. Inverse matrix calculation. Matrix diagonalization.
II Differential Calculus in ℝn
1. Dot Product Dot product. Euclidean spaces. Cauchy-Schwarz inequality. Orthogonal bases.
Projections. Gram-Schmidt orthogonalization process. Cross and mixed products properties and
geometrical applications
2. Topology & Scalar and Vector Fields - Notions of topology. Scalar and vector fields. Domain and range.
Graphical representation. Level sets of scalar fields.
3.Limits and Continuity - Limit in scalar and vector fields. Branching limits. Properties of limits. Continuity
and continuity prolongation.
4. Differential calculus -Differentiability of scalar and vector fields.
1. Vector spaces
2. Linear functions
3. Matrices and Linear Systems of Equations .
4. Determinants Permutations.
5. Eigenvalues and eigenvectors Definitions. The caracteristic polynomial. Algebraic and geometric
multiplicities. Inverse matrix calculation. Matrix diagonalization.
II Differential Calculus in ℝn
1. Dot Product Dot product. Euclidean spaces. Cauchy-Schwarz inequality. Orthogonal bases.
Projections. Gram-Schmidt orthogonalization process. Cross and mixed products properties and
geometrical applications
2. Topology & Scalar and Vector Fields - Notions of topology. Scalar and vector fields. Domain and range.
Graphical representation. Level sets of scalar fields.
3.Limits and Continuity - Limit in scalar and vector fields. Branching limits. Properties of limits. Continuity
and continuity prolongation.
4. Differential calculus -Differentiability of scalar and vector fields.
Teaching Methods
The teaching of this course comprises theoretical (3h/week), practical classes (2h/week), and tutorial hours
(per 2.5h/week):
Evaluation comprises:
Continuous assessment composed of two moments of evaluation, each for the topics and contents
lectrured previously from the beginning of the course and from the first assessment until the end of the
course. Students are passed with in both assessments have a mean grade larger or equal to 10.
Exam written test comprising all the contents lectured during the course. Students are passed if have a
grade larger or equal to 10.
Recursion Exam Assessment of all the contents lectured during the course. All students that failed in the
continuous assessment or in the exam are automatically registered into this exam, as well as, all the
students did not made any previous assessment. Students are passed if have a grade larger or equal to 10.
(per 2.5h/week):
Evaluation comprises:
Continuous assessment composed of two moments of evaluation, each for the topics and contents
lectrured previously from the beginning of the course and from the first assessment until the end of the
course. Students are passed with in both assessments have a mean grade larger or equal to 10.
Exam written test comprising all the contents lectured during the course. Students are passed if have a
grade larger or equal to 10.
Recursion Exam Assessment of all the contents lectured during the course. All students that failed in the
continuous assessment or in the exam are automatically registered into this exam, as well as, all the
students did not made any previous assessment. Students are passed if have a grade larger or equal to 10.
Teaching Staff (2023/2024 )
- Bruno Miguel Antunes Dinis [responsible]
- Luís Manuel Balsa Bicho
