Overview
The Faculty of Civil Engineering at the Czech Technical University in Prague (CTU) offers a comprehensive Calculus II course as a core component of its undergraduate engineering curricula. Designed for secondyear students, the course builds on the foundational concepts introduced in Calculus I, extending them to cover techniques and applications that are essential for advanced engineering analysis.
Learning Objectives
By the end of the semester, students are expected to:
- Master techniques for integrating rational, trigonometric, and improper functions.
- Apply series expansions to solve differential equations and evaluate limits.
- Understand and compute multivariate integrals, including double and triple integrals.
- Use polar, cylindrical, and spherical coordinates to simplify complex integrations.
- Analyze the convergence of infinite series and power series, and apply them in engineering contexts.
- Develop confidence in modeling physical phenomena such as heat flow, fluid dynamics, and structural deformation using calculus tools.
Course Structure
The semester is divided into twelve twohour lectures per week, supplemented by a weekly tutorial session where students practice problemsolving under the guidance of teaching assistants. The timetable typically follows this pattern:
- Weeks 13: Techniques of integration substitution, integration by parts, partial fractions.
- Weeks 46: Improper integrals and applications to probability.
- Weeks 79: Series and sequences convergence tests, Taylor and Fourier series.
- Weeks 1012: Multivariable calculus double and triple integrals, change of variables.
- Weeks 1314: Vector fields and the theorems of Green, Gauss, and Stokes.
- Week 15: Review and final exam preparation.
Key Topics
Integration Techniques
Beyond the basic methods reviewed in Calculus I, students explore advanced strategies such as trigonometric substitution, integration of rational functions using partial fractions, and the method of undetermined coefficients for evaluating definite integrals that arise in engineering problems.
Improper Integrals
The course examines integrals with infinite limits or unbounded integrands, focusing on convergence criteria and applications to realworld phenomena like heat transfer and signal processing.
Infinite Series
Students study the behavior of series, learning comparison, ratio, and root tests. Power series expansions of elementary functions are used to approximate solutions of differential equations, while Fourier series introduce students to the analysis of periodic signals.
Multivariable Calculus
The transition to multiple dimensions includes:
- Evaluation of double and triple integrals over general regions.
- Use of Jacobian determinants for coordinate transformations.
- Application of the divergence theorem, Green's theorem, and Stokes' theorem to relate integrals over domains to integrals over their boundaries.
Vector Calculus in Engineering
Realworld case studies illustrate how vector fields model fluid flow, electromagnetic fields, and stress distribution. Theorems from vector calculus provide powerful tools for simplifying calculations in these contexts.
Assessment and Grading
The final grade is a weighted combination of three components:
- Homework assignments (30%): Weekly problem sets focusing on the current topic.
- Midterm examination (30%): A 60minute written test covering material from weeks 18.
- Final examination (40%): A comprehensive 120minute exam that tests both theoretical understanding and practical problemsolving skills.
All assessments are conducted in Czech, but the majority of mathematical notation and formulas are universally understood. Students may submit written work in either Czech or English where appropriate.
Learning Resources
CTU provides a range of resources to support student success in Calculus II:
- Course Textbook: Advanced Calculus for Engineers by K. evk (in Czech) the primary reference for lecture material and exercises.
- Online Platform: The universitys Moodle system hosts lecture slides, supplemental videos, and a forum for peertopeer discussion.
- Study Sessions: Weekly review workshops run by senior students, held in the Faculty Library.
- Mathematical Software: Access to MATLAB and Mathematica licenses for visualizing multivariable functions and performing numerical integration.
Why Calculus II Matters for Engineers
Calculus II equips future engineers with analytical tools that are indispensable in design, simulation, and research. Whether a student intends to specialize in structural mechanics, environmental engineering, or electrical power systems, the ability to model continuous phenomena and evaluate complex integrals directly impacts the quality and safety of engineered solutions.
Moreover, the rigorous prooforiented approach nurtured in the course strengthens logical reasoning, a skill that is valued by employers across industries. Graduates of CTU who have mastered Calculus II often find themselves better prepared for graduate studies and for tackling interdisciplinary challenges.
How to Enroll
Enrollment is automatic for students pursuing a Bachelors degree in any engineering discipline at CTU. For exchange students or those entering a masters track, the course can be selected through the universitys online registration portal. Prerequisite verification ensures that every participant has successfully completed Calculus I with a minimum grade of satisfactory.
For additional information, prospective students may contact the Facultys Academic Office via studijni-oddeleni@fel.cvut.cz or visit the CTU Faculty of Civil Engineering website.
Student Testimonials
"The multivariable integration lectures opened my eyes to how we can simplify seemingly impossible problems. The examples from fluid dynamics made the theory feel very concrete." Jana K., 2ndyear civil engineering student.
"I appreciated the series of tutorial sessions where we could work through challenging proofs together. It helped me gain confidence for the final exam." Tom L., 2ndyear mechanical engineering student.
