Mathematics for Scientists and Engineers

Overview

Vectors: Vectors in the plane and space. Coordinates, scalar product, projections, and curl product.

Complex numbers: Concept of complex plane, vectorial and exponential representation of complex numbers. Fundamental operations with complex numbers: sum, subtraction, product, division, power and roots, and complex conjugate, Euler and de Moivre’s theorems

Fundaments of trigonometry: Sine, cosine, tangent functions. Their graphs in one dimensions, their representation on the unitary sphere, and representation as complex exponentials.

Elements of linear algebra: Definition of matrices and operations. Determinant of a matrix. Solution of a system of linear equations. Gauss’ elimination method. Eigenvalues/eigenvectors. Definition and basic properties of a vectorial space, isomorphisms and homomorphisms. Generalised definition of norm and scalar product.

Elements of Euclidean geometry: equation of a line and a plane. Equation of the circle and the ellipse.

Analysis of a single-variable function: Definition of a function. Definition of limit and derivative. Methods to calculate limits and derivatives. Definition of continuity and singularities. Study of a function.

Taylor and MacLaurin series and approximation of single-variable function: definition of orders of expansion

Integration in one variable: definition of definite and indefinite integral, integration by parts and by substitution, integral of a rational function, Gaussian integrals.

Ordinary differential equations: Definition of linearity and order of differential equations. Solutions for linear differential equations and main properties. Solution of specific non-linear cases.

Probability distributions: Probability concepts. Binomial, poisson and normal distributions.

Elements of discrete calculus: Series with their limit and convergence theorems and methods.

Learning Objectives

Display knowledge of, and apply practically, a range of mathematical techniques and properties in the areas of trigonometry, Euclidean geometry, probability, vectors, linear algebra, complex numbers, and single and multi-variable calculus.

Formulate mathematical problems and obtain analytical or approximate solutions.

Skills

Problem solving. Communicating mathematical concepts in a clear and concise manner both orally and in written form. Working independently and meeting deadlines.

Assessment

Examination must be passed