Mathematics (46-60)

Aim

After completing this course with a passing grade the student should be able to • know and be able to use, explain and apply fundamental concepts and methods of calculus of several variables;• demonstrate basic skills in calculus, problem solving and applications of calculus of several variables by applying its fundamental concepts , theorems and methods;• to perform standard calculations in calculus of several variables with good security;• perform didactic analysis of key concepts, operations , theorems and methods in algebra and statistics;• demonstrate the understanding of mathematical proofs and analyze how intuitive and logical thinking can complement each other to the understanding of mathematical concepts and methods;• discuss students' perceptions of and ways to acquire basic mathematical concepts and skills in algebra, probability and statistics based on a critical review of the research literature;• describe, analyze and reflect on various aspects of mathematical problem solving and its impact on teaching and learning mathematics;• describe and critically analyze various approaches to interdisciplinary and project-oriented work in school mathematics;• conduct, report and evaluate a larger mathematical problem-solving and project exercises

Content

Euclidean n-space. Basic topological concepts of the Euclidean n-space. Functions of several variables and vector-valued functions, including the following concepts: Graph, level curve, level surface. Limits and continuity, differentiability, partial derivatives, differentials, the chain rule. Gradient and directional derivative. Tangent and normal planes, linear approximation. Taylor’s Formula. Extreme-value problems. Invertibility and implicitly defined functions. Jacobian matrix and Jacobian determinant. Coordinate changes. Multiple integrals and iterated integral. Change of variables for multiple integrals: polar and spherical coordinates. Applications: area, volume, moments and center of mass. Mathematical reasoning, the mathematical proof and didactic analysis in connection to the relevant concepts and skills in algebra, calculus, probability and statistics related to mathematics education research.The course also discusses various approaches to and aspects of mathematical problem solving and project work at the school level in relation to a relevant mathematics education research on problem solving and project work.