At the end of this course the student should be able to:
- Have a good understanding of the knowledge of the basic applied mathematics for engineers, within the wide area of the differential and integral calculus of many variables, as well as of the vector analysis, which is adequate to his/her science.
- Know the new notions in the form of definitions and theorems that concern the basic contents of the course "Multivariable Calculus and Vector Analysis", in order to be able to apply them.
- Combine and make worthy of the knowledge that he/she acquired to other fields of the theoretical and applied mathematics, in which certain notions and principles of the present course are necessary and useful.
At the end of the course the student will have further developed the following skills and competences:
- Ability to demonstrate knowledge and understanding of essential concepts, principles and applications that are related to the differential and integral calculus of many variables, as well as to the vector analysis.
- Ability to apply such knowledge to the solution of problems in other fields of the wide conception of theoretical and applied mathematics, related to the science of Chemical Engineering, or to the solution of multidisciplinary problems.
- Study skills needed for continuing profession development.
There are no prerequisite courses. It is, however, recommended that students should have the basic knowledge of the differential and integral calculus of one variable, as well as of the linear algebra, which they were taught to the corresponding course "Single Variable Calculus and Linear Algebra".
Cartesian, cylindrical and spherical coordinates in space. Cylindrical surfaces and second degree surfaces. Functions of many variables, limit, continuity, partial derivative of first or higher order of functions and geometrical meaning. Derivation rules, Schwartz’s theorem and directional derivative. Total differential and the conception of differentiation. Composite functions and homogeneous equations, complex forms and basic existence theorems. Jacobian determinant and functional dependence. Taylor’s and Maclaurin’s mean value theorems. Extremities of functions and bounded extremities, Lagrange’s multipliers. Vector analysis and vectors in space. Limit, continuity and derivative of vector functions of one and many variables. Elements of the differential geometry of curves in space. Position vector of particle, vector velocity and acceleration. Unit tangential and unit perpendicular vector of curve. Orthogonal coordinate system or trihedral Frenet-Serret, curvature and turning of curve. Gradient or grade of functions, tensor product, divergence and rotation or swirling of vector functions, their physical meaning and basic vector identities. Laplace’s differential operator, harmonic functions and partial differential equations of Helmholtz, wave and diffusion. Irrotational and solenoidal fields, Helmholtz’s decomposition theorem. Curvilinear coordinate systems, vector meaning of Jacobian determinant, special orthogonal and curvilinear coordinates, transformations and change of coordinates. Applications of partial derivatives to geometry, tangential plane and perpendicular straight line to surface, tangential straight line and perpendicular plane to curve. Multiple integration of functions, double and triple integrals, change of coordinate system and applications to the calculation of plane surface areas, of volumes of three-dimensional domains, of mass, of moments of inertia and of gravity center. Curve integrals of the first and of the second kind, application to the calculation of the force work and Green’s theorem for the plane. The meaning of the circulation of vector functions, curve integrals independent of the root of integration and applications. Surface integrals and surface parameterization, application to the calculation of the area of arbitrary surface in space. Gauss’ and Stokes’ or Green’s for the space integral theorems and their physical meaning.
Use of web pages and e–class.
- Teaching (4 hours/week): lectures using blackboard of the theory and its application to typical mathematical problems of Chemical Engineering.
- Recitation (2 hours/week): solving on the blackboard exercises concerning mainly mathematical applications of the science of Chemical Engineering.
Total Module Workload (ECTS Standards):
The instruction language is Greek. Written or / and oral examination (100% of final mark).
- P.M. Hatzikonstantinou, "Mathematical Methods for Engineers and Scientists: Calculus of Functions with Many Variables & Vector Analysis", Gotsis Publications, Patras, 2017 (Eudoxos / code 68381163).
- J. Hass, C. Heil and M.D. Weir, "Thomas Infinitesimal Calculus" (translation Y. Kotsopoulos), Institute of Technology & Research – University of Crete Publications, Herakleion, 2018 (Eudoxos / code 77107082).
- D. Georgiou, T. Karakasidis and A. Megaritis, "Differential and Integral Calculus of Functions with Many Variables", Tziolas Publications, Thessaloniki, 2022 (Eudoxos / code 112690385).