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Fundamentals of Mathematics II

Code: 106551
Credits: 9
2026/2027
Degree programme Type Course
Bachelor in Artificial Intelligence FB 1

Contact lecturer

Name :
Alberto Dayan
Email :
alberto.dayan@uab.cat

Group languages

You can consult this information at the end of the document.

Prerequisites

Fonaments I and basic notions of calculus in one variable, such as continuity, differentiability and integration.

Objectives

The course contains three fundamental parts: differential and integral calculus in several variables, and vector analysis.


The objectives of the course are:


(i) Understand the basic concepts in each of these parts. These concepts include both the definitions of the mathematical objects being introduced and their interrelationship.


(ii) To know how to apply the concepts studied in a coherent way to the approach and resolution of problems.


(iii) Acquire skills in mathematical writing and calculus.

Learning outcomes

  • KM02 (Explain the concepts of differential and integral calculus needed to develop learning algorithms.) Explain the concepts of differential and integral calculus needed to develop learning algorithms.
  • KM03 (Describe the function optimisation methods used for training learning models.) Describe the function optimisation methods used for training learning models.
  • SM01 (Apply concepts of linear algebra, calculus, probability, and statistics to problem solving in the context of artificial intelligence applications.) Apply concepts of linear algebra, calculus, probability, and statistics to problem solving in the context of artificial intelligence applications.
  • SM02 (Interpret the mathematical formulation associated with algorithms and procedures in the field of artificial intelligence.) Interpret the mathematical formulation associated with algorithms and procedures in the field of artificial intelligence.
  • SM03 (Appropriately use mathematical language to formulate solutions to problems that require the use of mathematical concepts in the context of artificial intelligence.) Appropriately use mathematical language to formulate solutions to problems that require the use of mathematical concepts in the context of artificial intelligence.
  • SM04 (Use computer tools and programming languages for problem solving and manipulation of mathematical objects.) Use computer tools and programming languages for problem solving and manipulation of mathematical objects.

Contents

(1) Functions of several variables


-Geometry of the plane and space.
-Graph of a function, curves and level surfaces.
-Directional derivatives.
-Differentiability. Chain rule. Higher order derivatives. Absolute and relative extremes.
-Critical points, saddle points. Hessian criterion forrelative extremes. Lagrange multipliers for the calculation of absolute extremes.



(2) Multiple integrals.
-Integral iterations. Fubini's theorem.
-Variable change theorem. Polar, cylindrical and spherical coordinates. Calculation of masses and centers of mass.



(3) Integrals on curves and surfaces.
-Parameters and parameterized surfaces.
-Implicitly given surfaces. Vector tangent to a curve at a point. Tangent plane and normal vector to a surface.
-Length of a curve. Area ofa surface. Line integrals.
-Flow of a vector field.



(4) Continuous optimization
-Optimization using gradient descent.
-Constrained optimization and Lagrange multipliers.
-Convex optimization.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Study 85 3.4 KM02, KM03, SM01, SM02
Practical sessions 10 0.4 SM04
Problems 35 1.4 SM01, SM03
Theory 40 1.6 KM02, KM03, SM01, SM02


The methodology will be the standard for this type of subject with theory classes, problems and practical sessions.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Exams 80% 5 0.2 KM02, KM03, SM01, SM02, SM03
Exercise practices 20% 50 2 SM04

Assessment consists of a first midterm exam, which will account for 40% of the final grade, and a second midterm exam, which will account for 40% of the final grade. The remaining 20% comes from the practical sessions.


There will be a make-up exam at the end of the course, during which students may retake either one or both midterms. The grade earned on the make-up exam replaces the previously earned grade.


To pass the course, the average of the corresponding grades must be 5 or higher, and the grade on each midterm must be 2.5 or higher.


Bibliography

M.P. Deisenroth, A.A. Faisal and C.S. Ong, Mathematics for maching learning, Cambridge University Press, 2020.


B. Demidovich. Problemas y ejercicios de Análisis Matemático. Ed. Paraninfo, 1970.


J. E. Marsden y A.J. Tromba. Cálculo vectorial, cuarta edición. Addison-Wesley Longman, 1998.


S. L. Salas y E. Hille. Calculus, Vol. 1 y 2, tercera edición. Reverté, Barcelona, 1995.

Software

In the exams we will let the students write in the language that be most comfortable for them, but in principle 
we prefer that they use the English. We will work within sage.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 71 English second semester afternoon
(PAUL) Classroom practices 711 English second semester afternoon