After completing this course, students should have developed a clear understanding of the fundamental concepts of multivariable calculus and a range of skills allowing them to work effectively with the concepts.
The basic concepts are:
Derivatives as rates of change, computed as a limit of ratios
Integrals as a 'sum,' computed as a limit of Riemann sums
The skills include:
Fluency with vector operations, including vector proofs and the ability to translate back and forth among the various ways to describe geometric properties, namely, in pictures, in words, in vector notation, and in coordinate notation.
Fluency with matrix algebra, including the ability to put systems of linear equation in matrix format and solve them using matrix multiplication and the matrix inverse.
An understanding of a parametric curve as a trajectory described by a position vector; the ability to find parametric equations of a curve and to compute its velocity and acceleration vectors.
A comprehensive understanding of the gradient, including its relationship to level curves (or surfaces), directional derivatives, and linear approximation.
The ability to compute derivatives using the chain rule or total differentials.
The ability to set up and solve optimization problems involving several variables, with or without constraints.
An understanding of line integrals for work and flux, surface integrals for flux, general surface integrals and volume integrals. Also, an understanding of the physical interpretation of these integrals.
The ability to set up and compute multiple integrals in rectangular, polar, cylindrical and spherical coordinates.
The ability to change variables in multiple integrals.
An understanding of the major theorems (Green's, Stokes', Gauss') of the course and of some physical applications of these theorems.
Course Content (Syllabus)
Multivariable calculus, Integration, Vector Analysis and applications
Course Bibliography (Eudoxus)
Βιβλίο : ΛΟΓΙΣΜΟΣ ΙΙ, ΦΙΛΙΠΠΟΣ Ι. ΞΕΝΟΣ Λεπτομέρειες
Βιβλίο : Λογισμός Συναρτήσεων Πολλών Μεταβλητών και Εσαγωγή στις Διαφορικές Εξισώσεις, Παπασχοινόπουλος Γ. - Σχοινάς Χ. - Μυλωνάς Ν. Λεπτομέρειες
Βιβλίο : ΔΙΑΝΥΣΜΑΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, MARSDEN J., TROMBA A. Λεπτομέρειες
Βιβλίο : Διαφορικός και ολοκληρωτικός λογισμός συναρτήσεων πολλών μεταβλητών, Μυλωνάς Νίκος Λεπτομέρειες