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School of Chemical and Environmental Engineering

Now offering two distinct diplomas: Chemical Engineering and Environmental Engineering

Differential and Integral Calculus II


School Environmental Engineering
Course Level Undergraduate
Course ID MATH 102 Semester 2nd
Course Category Required
Course Modules Instruction Hours per Week ECTS
Lectures and Tutorial Exercises 4
Th=3, E=1, L=0
Course Type  General Background
Instruction/Exam Language Greek
The course is offered to Erasmus students No
Course URL https//   (in Greek)



Learning Outcomes

This course is continuation of Math 101 and aims to provides the essential mathematical techniques that an engineer needs in the theoretical part of his training as well as  what he/she  shall need in the computational part of his work.

On completion of this course students will be expected to

  • understand the basic vector operations, addition multiplication of vectors, find the projection of a vector on another vector
  • find areas of parallelograms and  volume of parallelepipeds  produced by vectors.
  • find the equations of lines and planes in space
  • calculate the partial derivatives of a function of 2or 3 variables
  • calculate the partial derivatives of functions given in implicit form
  • find gradient vector and the directional derivative of functions of two or three  variables
  • find whether a vector field is the gradient of a real valued function
  • find the  extreme values of functions of two variables  using the criterion of the second partial derivative
  • find the extreme values of functions under constrains using the Lagrange method
  • evaluate the volumes and areas of solids with the useage of double and triple integrals
  • find the work of force in a curve and examine whether the force is conservative
  • find the flow of a fluid  along or across a plane curve
  • evaluate the divergence and rotation of a vector field and their relation with the flow  of fluid  across  along or across a plane curve  (Green’s theorem), the boundary of a surface (Stoke’s theorem) or along a closed surface (Gauss theorem)
General Competencies/Skills
  • Review, analyse and synthesise data and information.
  • Work autonomously.
  • Work in teams.
  • Work in an international frame.


Vectors in Two and three Dimensional space. Inner, Cross, triple product. Lines and planes in Space. Polar, Cylindrical and Spherical Coordinates. Curves in the space and their Tangents. Integrals and derivatives of Vector Functions. Velocity, Tangential and Normal components of Acceleration. Functions of two and more variables. Partial Derivatives of functions of several variables, div, grad, Curl. The Chain rule. Directional Derivatives and Gradient vectors. Tangent Planes and differentials. Extreme values and Saddle points. Lagrange multipliers. Line Integrals. Path independence, Conservative Fields. Double and Triple integrals. Applications in Physics and Geometry :Moments and Centers of Mass, Volumes of Solids. Green’s theorem in the plane. Surface Integrals. Stoke’s Theorem. The divergence (Gauss’s) Theorem.


Lecture Method Direct (face to face)
Use of Information and Communication Technology E-class support
Instruction Organisation Activity Workload per Semester
- Lectures 39
- Tutorial Exercises 13
- Autonomous study 73
Course Total 125

Assessment Method

Ι. Written final examination (100%)


  • THOMAS Calculus ,  George B. Thomas , Jr., Joel Hass, Christopher Heil, Maurice D. Weir
  • Calculus Briggs William, CochranLyle, Gillett Bernard


Course Instructor: Associate Professor A. Manoussakis (Faculty - EnvEng)
Lectures: Associate Professor A. Manoussakis (Faculty - EnvEng)
Tutorial exercises: Associate Professor A. Manoussakis (Faculty - EnvEng)
Laboratory Exercises: