Mathematics II

In the first part of the course, students aim to extend their fundamental knowledge of Differential and Integral Calculus to real functions of n independent variables and apply these concepts to various scientific fields in the Natural Sciences (Chemistry, Physics, etc.). The second part of the course introduces students to the basic concepts of Probability Theory, random variables and distributions, as well as the fundamental principles of Statistics and Quantitative Analysis. Students are expected to be able to use methods for data description and exploratory data analysis, analyze correlations and agreements, and model relationships using regression methods.

Upon successful completion of this course, students will be able to:

• Understand and apply the fundamental concepts of vectors, operations in the n-dimensional Euclidean space, and geometric interpretations such as projections and inner, outer, and mixed products.
• Describe and analyze real functions of several variables, including limits, continuity, and graphical representation in multiple dimensions.
• Compute and interpret partial derivatives of multivariable functions, understand their geometric meaning (gradient, tangent plane), and extend differentiation to higher-order partial derivatives.
• Identify and determine extrema of functions of several variables and generalize optimization techniques.
• Apply integral calculus to evaluate double integrals over rectangular and general regions and interpret their applications in physical and chemical contexts.
• Understand the basic concepts of Probability Theory, including counting principles and combinatorics (permutations, arrangements, combinations).
• Apply the fundamental theorems of probability, including conditional probability, Bayes’ theorem, and the laws of multiplication, joint, and marginal probabilities.
• Recognize and use discrete and continuous probability distributions, including Bernoulli, Binomial, Geometric, Hypergeometric, Poisson, Uniform, Normal, and Exponential distributions.
• Use graphical and numerical methods for data summarization and exploratory data analysis (bar charts, pie charts, histograms, boxplots).
• Apply descriptive statistics to compute and interpret measures of central tendency, relative position, variability, skewness, and kurtosis for grouped and ungrouped data.
• Perform correlation analysis using parametric and non-parametric correlation coefficients, and assess agreement between qualitative (Cohen’s/Fleiss’ Kappa) and quantitative variables (Lin’s Concordance Coefficient, Kendall’s W, Bland–Altman method).
• Construct and evaluate linear regression models using the least squares method and assess goodness of fit.
• Develop analytical and quantitative reasoning skills for solving problems in scientific disciplines through the use of calculus, probability, and statistics.