Vector Calculus, Linear Algebra and Analytic Geometry
| Title | ΔΙΑΝΥΣΜΑΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, ΓΡΑΜΜΙΚΗ ΑΛΓΕΒΡΑ ΚΑΙ ΑΝΑΛΥΤΙΚΗ ΓΕΩΜΕΤΡΙΑ / Vector Calculus, Linear Algebra and Analytic Geometry |
| Code | ΜΑΥ1202 |
| Faculty | Sciences |
| School | Physics |
| Cycle / Level | 1st / Undergraduate |
| Teaching Period | Winter/Spring |
| Coordinator | Efthymia Meletlidou |
| Common | No |
| Status | Active |
| Course ID | 600021658 |
Programme of Study: PROGRAMMA SPOUDŌN 2022
Registered students: 0
| Orientation | Attendance Type | Semester | Year | ECTS |
|---|---|---|---|---|
| KORMOS | Compulsory Course | 1 | 1 | 8 |
| Academic Year | 2026 – 2027 |
| Class Period | Winter |
| Faculty Instructors |
|
| Weekly Hours | 5 |
| Class ID | 600306958
|
Type Of Offer
- Disciplinary Course
Course Type 2021
General Foundation
Mode of Delivery
- Face to face
Digital Course Content
- e-Study Guide https://qa.auth.gr/en/class/1/600306958
- At the Website of the School: https://www.physics.auth.gr/courses/10
- eLearning (Moodle): https://elearning.auth.gr/course/view.php?id=6235
Language of Instruction
- Greek (Instruction, Examination)
Prerequisites
General Prerequisites
Basic mathematical concepts and techniques acquired during previous levels of education. In particular, students are expected to be familiar with vectors, Cartesian coordinate systems, and fundamental geometric concepts. In addition, a basic understanding of complex numbers is required for the full comprehension of the course content; such knowledge is introduced during the first semester of studies
Learning Outcomes
- Understanding of the fundamental concepts of vector calculus, including familiarity with the geometric interpretation of vectors in three-dimensional space.
- Familiarity with the core concepts of linear algebra, such as matrices, determinants, and systems of linear equations.
- Understanding of eigenvalues and eigenvectors, as well as their significance in the theory of linear transformations.
- Familiarity with essential concepts of analytic geometry in space, including the representation of lines, planes, and conic sections, as well as the use of various coordinate systems (Cartesian, polar, cylindrical, spherical).
- Strengthening the ability to connect algebraic and geometric concepts, for the purpose of solving problems related to spatial structures and representations.
General Competences
- Apply knowledge in practice
- Retrieve, analyse and synthesise data and information, with the use of necessary technologies
- Work autonomously
- Be critical and self-critical
- Advance free, creative and causative thinking
Course Content (Syllabus)
-Introduction to Vector Calculus: Vector addition and subtraction. Scalar multiplication of a vector. Algebraic form of a vector. Dot product of vectors.
-Structures of Vector Spaces: Basic algebraic structures. Vector spaces. Subspaces.
-Linear Dependence and Bases: Linear combinations of vectors. Linear dependence and independence. Basis and dimension of a vector space.
-Vector Representation and Geometric Concepts: Coordinates in rectangular and oblique systems. Vector projection. Direction cosines. Cross, triple, and double cross products.
-Matrices and Their Properties: Definition of a matrix. Matrix operations. Symmetric, skew-symmetric, and orthogonal matrices. Matrix powers. Complex matrices. Generalization of inner product for real and complex vectors.
-Determinants and Inverse Matrices: Properties of determinants. Determinant computation. Inverse matrix using determinants.
-Matrix Transformations and Linear Systems of Equations: Elementary matrix transformations. Row echelon forms. Rank of a matrix. Homogeneous and non-homogeneous linear systems of equations.
-Methods for Solving Linear Systems of Equations: Gaussian elimination. Cramer’s rule. Inverse matrix method. Solution and analysis of linear systems.
-Eigenvalues and Eigenvectors: Definitions and properties. Computation methods. Cayley–Hamilton theorem. Similar matrices and similarity transformations. Matrix diagonalization. Minimal polynomial.
-Change of Basis and Coordinate Systems in Euclidean Space: Translation and rotation of axes. Cartesian, polar, cylindrical, and spherical coordinate systems. Transformations between coordinate systems.
-Lines and Planes in Euclidean Space: Equations of lines and planes. Relative positions of lines and planes. Position of a line relative to a plane.
-Conic Sections: Circle. Parabola. Ellipse. Hyperbola. Definitions and geometric properties.
Keywords
Vector calculus, Linear algebra, Analytic geometry, Vector spaces, Matrices, Determinants, Eigenvalues and eigenvectors, Coordinate systems, Conic sections, Linear systems
Educational Material Types
- Notes
- Slide presentations
- Book
Use of Information and Communication Technologies
Use of ICT
- Use of ICT in Course Teaching
- Use of ICT in Communication with Students
Description
Information and Communication Technologies (ICT) are used in the course for the teaching of concepts that require visual representation, primarily through PowerPoint presentations and the use of Mathematica software.
In cases where logistical or practical issues prevent the in-person delivery of the course, lectures are conducted remotely via the ZOOM teleconferencing platform.
Communication with students is carried out through the official AUTH eLearning platform, where notes and announcements are posted. Email communication is also used when necessary.
Course Organization
| Activities | Workload | ECTS | Individual | Teamwork | Erasmus |
|---|---|---|---|---|---|
| Lectures | 190 | 6.3 | ✓ | ||
| Exams | 5 | 0.2 | ✓ | ||
| Problem solving | 45 | 1.5 | ✓ | ||
| Total | 240 | 8 |
Student Assessment
Description
Student assessment is based on a final written examination. Additionally, a mid-semester optional written test (Progress Test) is offered around mid-November, covering the material of Vector Calculus. The maximum grade that can be obtained in the Progress Test is 2.5, while the minimum passing grade is 1.2. The Progress Test exempts students from the corresponding topic in the final exam, and its grade is included in the final score. If a student is not satisfied with their Progress Test score, they may choose to be re-examined on the relevant material during the final exam.
Student Assessment methods
- Written Exam with Problem Solving (Formative, Summative)
Bibliography
Course Bibliography (Eudoxus)
- Μια Εισαγωγή στη Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία, Γαϊτάνος Θ., Μελετλίδου Ε., Μουστακίδης Χ., Παπαδόπουλος Δ., Πασχάλης Ι., Εκδόσεις Ζήτη, 2019 (Εύδοξος: 77118476) ΔΗΛ
- Γραμμική Άλγεβρα, Αναλυτική Γεωμετρία και Εφαρμογές, Καδιανάκης Ν., Καρανάσιος Σ., Εκδόσεις Συμμετρία, 2017 (Εύδοξος: 68382505) ΔΗΛ
- Εισαγωγή στη Γραμμική Άλγεβρα και Αναλυτική Γεωμετρία (2η Έκδοση), Ιωαννίδου Θ., Εκδόσεις Τζιόλα, 2021 (Εύδοξος: 77106815) ΔΗΛ
Additional bibliography for study
- Διανυσματικός Διαφορικός Λογισμός [Κεφάλαιο]. Μπράτσος Α., Μαθήματα Ανώτερων Μαθηματικών, Κάλλιπος – Ανοικτές Ακαδημαϊκές Εκδόσεις, 2015.
- Μια Εισαγωγή στη Γραμμική Άλγεβρα [Κεφάλαιο]. Χαραλάμπους & Φωτιάδης, Μια Εισαγωγή στη Γραμμική Άλγεβρα, Κάλλιπος – Ανοικτές Ακαδημαϊκές Εκδόσεις, 2015.
- Στοιχεία Γραμμικής Άλγεβρας. Παπαϊωάννου Σ., Βογιατζή Δ., Πίνακες, Ορίζουσες και Γραμμικά Συστήματα, Κάλλιπος – Ανοικτές Ακαδημαϊκές Εκδόσεις, 2015
- Linear Algebra and Its Applications, Lay D. C., Pearson, 2015
- Introduction to Linear Algebra, Strang G., Wellesley-Cambridge Press, 2016
- Vector Calculus, Marsden J. E., Academic Press, 2003
- Linear Algebra Done Right, Axler S., Springer, 2015
Last Update
15-07-2025
