Enquiry for Science Major/Minor/Programme Requirements
MATH1851 Calculus and ordinary differential equations (6 credits) Academic Year 2026
Offering Department Mathematics Quota 700
Course Co-ordinator Dr V Bhat, Mathematics < vbhat@hku.hk >
Teachers Involved (Dr V Bhat,Mathematics)
(Prof K W Chow,Mechanical Engineering)
(Prof Y Wei,Mechanical Engineering)
Course Objectives In this course, students will be introduced to fundamental concepts of calculus and ordinary differential equations with a view on applications in different engineering fields. A concrete foundation of mathematics that underpins the various engineering subjects will be built. Mathematical concepts and principles, as well as some typical engineering applications, would be emphasized so that students could enhance their mathematical skills in solving engineering problems, and be well prepared in learning a higher level of applied mathematics required in different engineering disciplines.
Course Contents & Topics - Differential and integral calculus (single variable) [limits and continuity, derivatives, (higher-order) derivatives of elementary functions, derivatives by implicit differentiation, the mean value theorem, L'H\^{o}pital's rule, parametric representation of curves, polar coordinates, indefinite integrals, integration by parts, partial fractions decomposition, definite integrals, the fundamental theorem of calculus, and their applications]
- Ordinary differential equations [first order equations, integrating factors and linear equations, Bernoulli equations, separable equations, homogeneous equations, exact differential equations, higher-order homogeneous linear equations with constant coefficients, characteristic polynomials, methods of undetermined coefficients and variation of parameters, higher-order inhomogeneous linear ordinary differential equations, choice of particular solutions and physical implication of resonance, Cauchy-Euler equations, and their applications]
- Laplace transforms [Laplace transforms of elementary functions, inverse Laplace transforms, transforms of derivatives and integrals, derivatives of  Laplace transform, first and second shifting theorems, convolutions, partial fractions, solution of linear differential equations (initial value problems) using Laplace transforms]
Course Learning Outcomes
On successful completion of this course, students should be able to:

CLO 1 demonstrate knowledge and understanding of basic calculus and ordinary differential equations as well as their relationship with some typical physical/engineering applications: unerringly perform the calculation details for the solution, and accurately correlate the solution approach with the fundamental concepts involved
CLO 2 apply mathematical skills to model and solve some basic physical/engineering problems: analyze the given problem, identify the appropriate mathematical skills, articulate a convincing rationale for the approach used, clearly give the mathematical formulation, and correctly find the solution
CLO 3 understand well established methods to solve differential equations, and correlate qualitatively with potential applications in engineering topics like oscillations and electric circuits. Identify the occurrence of resonance where large amplitude displacements can be expected
CLO 4 explore the technique and usage of integral transform, using the Laplace transform as an illustrative example. Appreciate the power of these techniques in initial value problems and applications like vibrations and signal processing
CLO 5 be well prepared to cope with a higher level of engineering mathematics required in different engineering disciplines
Pre-requisites
(and Co-requisites and
Impermissible combinations)		 Level 2 or above in Module 1, or Module 2 of HKDSE Mathematics or equivalent, or
Pass in MATH1011; and
Not for students who have passed MATH1013 or MATH1861, or have already enrolled in these courses.

(This course is exclusively for Engineering students whose primary major or programme requires this as a core course.)
Course to PLO Mapping
Offer in 2026 - 2027 Y        1st sem    2nd sem    Examination Dec    May     
Offer in 2027 - 2028 Y
Course Grade A+ to F
Grade Descriptors
A Demonstrate an excellent understanding of key concepts and ideas by being able to identify the appropriate theorems and methods and their applications through correctly analysing problems, clearly and elegantly presenting correct logical reasoning and argumentation and being able to carry out computations carefully and correctly, and with some innovative approaches to solving problems.
B Demonstrate a good understanding of key concepts and ideas by being able to identify the appropriate theorems and methods and their applications through correctly analysing problems, but with some minor inadequacies in arguments, identifying the appropriate theorems and methods or their applications and presentation or with some minor computational errors.
C Demonstrate an acceptable understanding of key concepts and ideas by being able to correctly identify appropriate theorems and methods, but with some inadequacies in applying them through incorrectly analysing problems with poor argument and presentation or a number of minor computational errors.
D Demonstrate some understanding of key concepts and ideas by being able to correctly identify appropriate theorems and methods, but with substantial inadequacies in applying them through incorrectly analysing problems with poor argument or presentation or with substantial computational errors.
Fail Demonstrate poor and inadequate understanding by not being able to identify appropriate theorems and methods or their applications, and not being able to complete the solution.
Communication-intensive Course N
Course Type Lecture-based course
Course Teaching
& Learning Activities
Activities Details No. of Hours
Lectures 36.0
Tutorials 12.0
Reading / Self study 100.0
Assessment Methods
and Weighting
Methods Details Weighting in final
course grade (%)		 Assessment Methods
to CLO Mapping
Assignments 10.0 1,2,3,4,5
Examination 60.0 1,2,3,4,5
Test 2 tests 30.0 1,2,3,4,5
Required/recommended reading
and online materials		 (Textbook) Introduction to Calculus and Differential Equations (Pearson)
G.B. Thomas, et al.: Thomas' Calculus (Pearson Education, 2023, 15th ed.)
R.K. Nagle, et al.: Fundamentals of Differential Equations and Boundary Value Problems (Pearson Education, 2008, 5th ed.)
Course Website http://moodle.hku.hk/
Additional Course Information There will be no 'make-up' for a missed test or assignment under normal circumstances.
Students are advised not to take MATH1851 and MATH1853 together in the same semester.
This course is offered by the Department of Mathematics and the Faculty of Engineering.


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