Course: 2024/2025

Mathematics

(19758)

Upon successful completion of this course, students will be able to:
1. Grasp the fundamental building blocks of mathematical analysis, including real numbers, functions, limits, and continuity. Students will develop a solid understanding of domains, ranges, and elementary functions such as polynomials, exponential and trigonometric functions.
2. Develop proficiency in calculating derivatives and integrals. Students will learn to interpret derivatives as rates of change, apply derivatives to solve problems involving tangent lines and linear approximations, and utilize the Fundamental Theorem of Calculus for integration to solve problems related to areas and other applications.
3. Acquire skills to solve simple differential equations and analyze their solutions, understanding concepts like fixed points and stability. This includes applications to modeling problems in neuroscience.
4. Engage with higher-order derivatives, Taylor expansions, and numerical approximations to understand local and asymptotic behavior of functions, including extrema, concavity, and behavior at infinity.
5. Understand and apply concepts of linear algebra, including matrix operations, systems of linear equations, vector spaces, and linear transformations. Students will also master eigenvalues and eigenvectors, diagonalization, and applications to stability analysis.
6. Delve into complex numbers, exploring their algebraic properties, geometric interpretation, and applications. Understand and apply concepts of orthogonality, inner product spaces, and orthogonal bases, critical for various applications in mathematics and physics.
By the end of this course, students will have developed a comprehensive mathematical toolkit, enabling them to tackle complex problems in neuroscience. This will include a proficiency in visualizing mathematical concepts in multiple dimensions and applying mathematical theory to practical and theoretical problems.

Skills and learning outcomes

Description of contents: programme

1. Building blocks
- Introduction to real numbers
- Functions
- Basic definitions of functions: input, output, domain, range
- Limits, continuity
- Elementary functions (polynomials, exponential, trigonometric)
2. Derivatives
- What is the derivative? Rate of change
- Introduction to differential equations
- Properties of the derivative
- Tangent line
- Linear approximations
- Basic derivatives
3. Higher order derivatives
- Interpretation
- Taylor expansions
- Introduction to power series and Taylor series
- Numerical approximations
4. Local and asymptotic behavior of functions
- Relative extrema
- Concavity and convexity
- Limits at infinity, asymptotes
5. Integration
- Calculating areas: the Riemann integral
- Properties of the integral
- Fundamental Theorem of Calculus
- Basic integration methods
6. Differential equations
- Solutions of differential equations
- Fixed points, stability
- Linear stability analysis
7. Linear Functions in many variables
- Linear Differential Equations as motivation
- Systems of linear equations
- Linear Transformations
- Matrix representation
8. Vector spaces
- Vector spaces
- Bases and Linear Combinations
9. Matrix algebra
- Inverse
- Determinant
10. Eigenvalues and eigenvectors
- Definition
- Eigenvectors as a new basis
- Diagonalization
11. Linear differential equations
- Equilibrium points
- Linear stability analysis
12. Complex numbers
- Complex eigenvalues as motivation
- Binary representation
- Modulus, argument
- Geometric interpretation
- Polar representation
- Euler's formula
13. Orthogonality
- Inner product
- Norm, distance
- Orthogonal sets
- Orthogonal bases
14. Fourier series (optional)
- Introduction, motivation
- Periodic functions
- Derivation
- Fourier series as change of basis

Learning activities and methodology

The learning methodology will include:
- Attendance to master classes, in which core knowledge will be presented that the students must acquire. The recommended bibliography will facilitate the students' work.
- Resolution of exercises by the student that will serve as a self-evaluation method and to acquire the necessary skills.
- Exercise classes, in which problems proposed to the students are discussed.
- Partial exams.
- Final Exam.
- Tutorial sessions.
- The instructors may propose additional homework and activities.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Extraordinary call: regulations

Basic Bibliography

- Alan Garfinkel , Jane Shevtsov , Yina Guo. Modeling Life: The Mathematics of Biological Systems. Springer Cham. 2017
- David C. Lay. Linear Algebra and Its Applications (5th Edition). Pearson. 2016

- José A. Cuesta · Calculus I: Differential and Integral Calculus of a Single Variable : https://ocw.uc3m.es/pluginfile.php/5548/mod_page/content/12/CuestaCalculusOCW.pdf
- Silvanus P. Thompson · Calculus Made Easy : https://calculusmadeeasy.org/
- Wulfram Gerstner, Werner M. Kistler, Richard Naud and Liam Paninski · Neuronal Dynamics: From single neurons to networks and models of cognition : https://neuronaldynamics.epfl.ch/

(*) Access to some electronic resources may be restricted to members of the university community and require validation through Campus Global. If you try to connect from outside of the University you will need to set up a VPN

The course syllabus may change due academic events or other reasons.