Mathematics

The math and civil engineering faculties at TU Delft offer several free online courses in mathematics, probability and statistics.

The mathematics department at TU Delft offers several MOOCs on the EdX and OpenCourseWare platforms, as well as Jupyter books. You can access the whole course for free from the links below. The content of these courses are elaborated in the table below (check remarks):

• Pre-university calculus

• Calculus I

• Calculus II

• Linear Algebra I

• Linear Algebra II

• Math explained

• Probability and Statistics Applications for Civil Engineers

• Probability theory

• Statistics

The following topics on Mathematics are considered prerequisite knowledge for the civil engineering MSc-program:

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Pre-university calculus

Subject	Topic category / Learning objectives	Open Educational Resources[1]	Remarks
Pre-university calculus	Elementary functions (power functions, roots, polynomials, trigonometric functions, exponential and logarithmic functions) ➤ Student can: - Understand, visualize and manipulate different elementary functions, like powers, roots, polynomials, trigonometric functions, exponential and logarithmic functions.	- Pre-university calculus[8]	-Pre-university calculus: weeks 1 and 2
	Equations and inequalities involving these elementary functions ➤ Student can: - Understand, visualize and solve equations and inequalities involving these elementary functions.	- Pre-university calculus[8] - Equations and inequalities with examples[9]	-Pre-university calculus: weeks 3 and 4 - Make sure you check the examples
	Differentiation and derivatives of compositions of elementary functions ➤ Student can: - Understand the concept of differentiation and to calculate the derivatives of compositions of elementary functions.	- Pre-university calculus[8] -Differentiation rule (Chain rule) with examples[10]	- Pre-university calculus: week 5
	Integration and elementary integration techniques ➤ Student can: - Understand the concept of integration and to use some elementary integration techniques	- Pre-university calculus[8] - Integration techniques[11]	- Pre-university calculus: week 6
	Geometric objects in the plane, such as vectors, lines, circles and more general curves ➤ Student can: - Implement geometric tools like coordinates, vectors and parameterizations -Use those to describe curves in the plane and to compute geometric quantities likes angles and distances - Recognize applications, such as the description of dynamics of physical systems.	- Pre-university calculus[12]	- Pre-university calculus: week 7 - Note this is a different link than the previous topics
	Ordinary Differential Equations (ODEs), Partial Differential Equations (PDEs) ➤ Student can: - Understand the concepts of direction field, first order linear differential equations and partial differential equations, the swinging pendulum Newton’s law, and interpret the meaning of a differential equation	- Math explained[8]	- ODEs lecture
	Complex numbers ➤ Student can: - Understand the concept of complex numbers - Make basic operations with complex numbers - Recognize the applications of complex numbers	- Math explained[8]	- Complex numbers lecture
	Set theory ➤ Student can: - Understand the concept of sets, as well as complements and differences. - Make basic operations with sets, such as union and intersection.	- Math explained[8]	- Sets lecture
	Cylindrical and spherical coordinates ➤ Student can: - Convert from cylindrical to rectangular coordinates. - Convert from rectangular to cylindrical coordinates. - Convert from spherical to rectangular coordinates. - Convert from rectangular to spherical coordinates.	- Cylindrical and spherical coordinates[4]

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Calculus

Subject	Topic category / Learning objectives	Open Educational Resources[1]	Remarks
Calculus CTB1001-16 AESB1211 CTB2200 AESB2440	Basic integration (multiple integrals), and differentiation (derivatives, partial derivatives, numerical differentiation) ➤ Student can: - Visualize functions of two variables and read off qualitative information from graphs and contour plots. - Calculate (higher order) partial derivatives for a variety of multivariable functions. - Calculate and interpret directional derivatives and gradients of multivariable functions. - Find and classify critical points of functions of two variables and determine extreme values for functions of compact domains. - Evaluate double and triple integrals over general integration domains using various coordinates systems.	- Calculus I[12] -Calculus II[12]	Calculus I course: week 5 - Calculus II course: all modules
	Taylor polynomials ➤ Student can: - Approximate and find solutions of differential equations with Taylor polynomials.	Calculus I[12]	Calculus I course: weeks 5 and 6

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Linear Algebra

Subject	Topic category / Learning objectives	Open Educational Resources[1]	Remarks
Linear Algebra CTB1002	Vectors (calculations, the dot product, the cross product, lines and planes) ➤ Student can: - Apply the dot product and cross product. - Describe lines, planes and their intersections.	- Linear algebra I[12]	- Linear algebra I course: week 1
	Linear equations (systems of equations, structure of the solutions set) ➤ Student can: - Solve systems of linear equations and describe the solution set.	- Linear algebra I[12]	- Linear algebra I course: week 2
	Linear dependence (linear combinations and linear dependence) ➤ Student can: - Decide whether vectors are linear dependent or not.	- Linear algebra I[12]	- Linear algebra I course: week 3
	Linear subspaces (basis and coordinates, dimension and the rank theorem) ➤ Student can: - Recognize linear subspaces, describe elements of linear subspaces using bases and coordinates.	- Linear algebra I[12]	- Linear algebra I course: week 4
	Orthogonality (orthogonal sets, orthogonal projections, Gram-Schmidt algorithm, orthogonal complements and transposition) ➤ Student can: - Calculate projections and orthogonal decompositions of vectors.	- Linear algebra I[12]	- Linear algebra I course: week 5
	Least-square solutions ➤ Student can: - Find least-square solutions of a system of linear equations and apply it to regression	- Linear algebra I[12]	- Linear algebra I course: week 6
	Matrix algebra (sum, product, inverse, transpose and transformations) ➤ Student can: - Perform algebraic operations on matrices such as matrix multiplication and matrix inversion. - Recognize linear transformations, apply their properties and find the standard matrix.	- Linear algebra II[12]	- Linear algebra II course: weeks 1 and 2
	Determinants ➤ Student can: - Find the determinant of a matrix and apply properties of determinants in the context of algebra and geometry.	- Linear algebra II[12]	- Linear algebra II course: week 3
	Eigenvalues and eigenvectors ➤ Student can: - Find eigenvalues, eigenvectors and eigenspaces of a matrix.	- Linear algebra II[12]	- Linear algebra II course: week 4
	Diagonalization, similarity transformations and coordinate transformations ➤ Student can: - Diagonalize a matrix if possible and perform other similarity transformations. - Apply properties of symmetric matrices.	- Linear algebra II[12]	- Linear algebra II course: week 5
	LU decomposition, Gaussian elimination ➤ Student can: - Find the singular value decomposition of a matrix.	- Linear algebra II[12]	- Linear algebra II course: week 6
	Inner products ➤ Student can: - Understand the concepts of inner product spaces and norms, comprehending definitions, properties, and geometric interpretations - Differentiate between different types of inner products and norms, applying these concepts practically.	- Inner product and norm[3]

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Probability & Statistics

Subject	Topic category / Learning objectives	Open Educational Resources[1]	Remarks
Probability & Statistics CTB2200 AESB2440	Probability spaces and general concepts ➤ Student can: - Understand the concepts of events, probability function and conditional probability. - Understand and apply conditional probability, independence, total probability and Bayes’ rule	- Probability theory[13] - General probability concepts[14]	- Probability theory course: week 1 - Probability and statistics for engineers: part 1
	Discrete and continous random variables ➤ Student can: - Understand the differences between discrete and continuous random variables - Understand and apply the following discrete distributions: Bernoulli, geometric, binomial, Poisson. - Understand and apply the following continuous distributions: density function, exponential, Pareto, uniform, normal, and cumulative.	- Probability theory[13] - Discrete and continuous random variables[14]	- Probability theory course: weeks 2 and 3 - Probability and statistics for engineers: parts 2 and 3
	Multivariate random variables ➤ Student can: - Understand and apply the concepts of variance, covariance and correlation, independence and conditional expectation - Apply joint distribution and marginal distribution	- Probability theory[13] - Joint, Independence and Conditional distributions[14]	- Probability theory course: week 4 - Probability and statistics for engineers: part 5
	Perform computations with random variables ➤ Student can: - Perform computations with random discrete and continuous variables - Understand and apply Jensen’s inequality and extremes	- Probability and Statistics Applications for Civil Engineers[14] - Computations with random variables[2]	- Probability and Statistics Applications for Engineers course - A modern introduction to probability and statistics: Chapter 8
	Basis hypothesis testing (incl. t-test) ➤ Student can: - Understand how to perform a test in various settings	Statistics[13]	Statistics course: week 3

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Numerical mathematics

Subject	Topic category / Learning objectives	Open Educational Resources[1]	Remarks
Numerical mathematics CTB2400	Approximate first and higher derivatives using, but not limited to; central differences; forward differences; backward differences. ➤ Student can: - Estimate the truncation error by means of Taylor series - Understand the impact of measurement errors on approximations of derivatives	Numerical differentiation[6]	- Numerical Methods for Ordinary Differential Equations: Chapter 3 - Download the book
	Estimate the error ➤ Student can: - Estimate the error value for different nummerical methods used	Estimating errors[7]	- See chapters 1.4 to 1.6
	Approximate the solution to nonlinear (systems of) algebraic equations ➤ Student can: - Solve nonlinear equations by means of the following methods; bisection, Picard iteration, Newton-Rhapson and its variants - Define and solve nonlinear systems of equations by means of various iterative methods.	Non-linear equations[6]	- Numerical Methods for Ordinary Differential Equations: Chapter 4
	A suitable stopping criteria ➤ Student can: - Choose a stopping criteria according to a chosen nummerical method	Stopping criteria[7]	- Go through stopping criteria in chapter 4
	Numerical integration ➤ Student can: - Use numerical integration methods. - Deduce the truncation and rounding errors.	Numerical integration[6]	- Numerical Methods for Ordinary Differential Equations: Chapter 5
	Numerical time-integration methods ➤ Student can: - Understand and apply the concepts of Truncation error / Floating point arithmetic and stability	Numerical time integration of initial-value problems[6]	- Numerical Methods for Ordinary Differential Equations: Section 6.4
	Finite-difference method ➤ Student can: - Derive a finite-difference and show the order of the resulting finite difference - Solve boundary value problems using the finite-difference method - Discretize the Neumann boundary condition with a virtual point - Recognize upwind discretization	Finite-difference method for boundary-value problems[6]	- Numerical Methods for Ordinary Differential Equations: Chapter 7
	Combine a finite difference method with a numerical time-integration method ➤ Student can: - Reproduce and use the following methods for initial value problems and investigate the truncation and rounding error behavior: Euler forward, Euler backward, Modified Euler and Trapezium rule - Use the Runge-Kutta 4 method	The instationary heat equation[6]	- Numerical Methods for Ordinary Differential Equations: Chapter 8
	Select appropriate numerical methods ➤ Student can: - Choose the best fitting nummerical method to solve a problem	Numerical methods[7]	- See chapters 2.6, 3.8, 4.7, 5.8, 6.12, 7.10 and 8.4

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[1] (1,2,3,4,5)

This is an experimental Jupyter Book, part of an educational research project, made by staff and MSc students of TU Delft. The first two columns with required prior knowledge were defined by the admission committee Civil Engineering. Column 3 and 4 with Open Educational Resources (OER) is experimental. These OER materials are provided as a service. Although we did our best to collect OER that reflect the required knowledge as good as possible, based on surveys among students and discussion with staff members, unfortunately we can not give a guarantee that the quality of all material is good. Suggestions are welcome via email

[8] (1,2,3,4,5,6,7)

Course provided by TUDelft at OpenCourseWare. (last checked october 2023)

[9]

Page is from math.vel.cvut, https://math.fel.cvut.cz/en/mt/txtz/0/txe3za0a.htm (last checked october 2023)

[10]

Page is from geeksforgeeks, https://www.geeksforgeeks.org/derivatives-of-composite-functions/ (last checked october 2023)

[11]

Page is from khan academy, https://www.khanacademy.org/math/calculus-all-old/integration-techniques-calc (last checked october 2023)

[12] (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)

Course provided by DelftX at EdX. (last checked october 2023)

[4]

Strang, G, Herman E. (2023). 12.7: Cylindrical and Spherical Coordinates.

[3]

The bright side of mathematics (2022). Linear Algebra - Part 9 - Inner Product and Norm lecture video.

[13] (1,2,3,4)

Course provided by TU Delft via EdX. (last checked october 2023)

[14] (1,2,3,4)

Course created by TU Delft authors. (last checked october 2023)

[2]

Lopuhaa, HP, et al. (2005). A modern introduction to probability and statistics. Springer Texts in Statistics.

[6] (1,2,3,4,5,6)

Vuik, C., van Beek, P., Vermolen, F. van Kan, J. (2006). Numerical Methods for Differential Equations. TU Delft.

[7] (1,2,3)

C. Vuik, F.J. Vermolen, M.B. van Gijzen, M.J. Vuik (2015). Numerical Methods for Ordinary Differential Equations. TU Delft.