Calculus II

 • Euclidean space Rn. Neighborhoods. Classification of points of Rn . Open and closed sets. Sequences. Basic theorems. • Functions of several variables. Limit of a function. Operations with limits. Continuous function. Properties of continuous functions. Partial derivatives. Partial derivatives of higher order. Differentiable function. Total differentials. Differentials of higher orders. Differentiation of composite functions. Implicit functions. Jacobians. Transformations. Inverse transformations. Directional derivatives. The mean-valued theorem and Taylor’s theorem for several functions. Extreme values. Extreme values under constraints. • Double and triple integrals. •Scalar and vector fields. Gradient, Divergence and Curl. Line integral of first kind and second kind. Green’s formula. Surface integral of the first and second kind. Stokes’ theorem. Gauss’ theorem. Conservative field. Solenoidal field.

The course aims to teach theorems and rules, to develop critical and analytical thinking so that interdisciplinary problems could be modelled and solved with mathematical accuracy and discipline.

Upon successfully completing this course the student should:

• understand the fundamental concepts of function of several variables, such as the limit, the continuity, the partial derivative, the differential, synthesise and apply the properties of these concepts in the study of the extreme points of a real function of several variables.
• have knowledge of the theoretical background for the study of double, triple and generalized integral of a real function of several variables and be able to apply the methods for calculating these  integrals.
• have knowledge of the theory and methodology so as to calculate one line and surface integral, which can apply to problems such as area of a surface, work along a curve, etc.

The course aims at the acquisition of knowledge, ideas and skills that are to be implemented in other courses related to Informatics and Biomedicine.

• ΛΟΓΙΣΜΟΣ ΣΥΝΑΡΤΗΣΕΩΝ ΠΟΛΛΩΝ ΜΕΤΑΒΛΗΤΩΝ ΓΙΑ ΤΙΣ ΕΠΙΣΤΗΜΕΣ ΤΟΥ ΜΗΧΑΝΙΚΟΥ, Ν.ΚΑΔΙΑΝΑΚΗΣ-Σ.ΚΑΡΑΝΑΣΙΟΣ-Α.ΦΕΛΛΟΥΡΗΣ, ΤΣΟΤΡΑΣ ΑΝ ΑΘΑΝΑΣΙΟΣ, 1η/2015, ΑΘΗΝΑ, 68379699
• Εφαρμοσμένος Διανυσματικός Απειροστικός Λογισμός – Β’ Έκδοση, ΛΕΩΝΙΔΑΣ Ν. ΤΣΙΤΣΑΣ, “Μ.ΑΘΑΝΑΣΟΠΟΥΛΟΥ-Σ.ΑΘΑΝΑΣΟΠΟΥΛΟΣ Ο.Ε.”, 2η έκδ./2003, ΑΘΗΝΑ, 45391
• Μαθηματική Ανάλυση ΙΙ,  Ρασσιάς Θ., ΤΣΟΤΡΑΣ ΑΝ ΑΘΑΝΑΣΙΟΣ, 1η/2014, ΑΘΗΝΑ, 41955064
• ΛΟΓΙΣΜΟΣ ΣΥΝΑΡΤΗΣΕΩΝ ΠΟΛΛΩΝ ΜΕΤΑΒΛΗΤΩΝ ΚΑΙ ΔΙΑΝΥΣΜΑΤΙΚΗ ΑΝΑΛΥΣΗ, ΚΩΝΣΤΑΝΤΙΝΙΔΟΥ ΜΑΡΙΑ, ΣΕΡΑΦΕΙΜΙΔΗΣ ΚΑΡΟΛΟΣ, “””σοφία”” Ανώνυμη Εκδοτική & Εμπορική Εταιρεία”,  1η/2012, ΘΕΣ/ΝΙΚΗ, 22766838
• Εφαρμοσμένη Ανάλυση και Θεωρία Fourier,  Φιλιππάκης Ε. Μιχαήλ, ΤΣΟΤΡΑΣ ΑΝ ΑΘΑΝΑΣΙΟΣ, 1η/2014, ΑΘΗΝΑ, 68403139
• THOMAS ΑΠΕΙΡΟΣΤΙΚΟΣ ΛΟΓΙΣΜΟΣ, [George B. Thomas], Jr., Joel Hass, Christopher Heil, Maurice D. Weir, ΙΔΡΥΜΑ ΤΕΧΝΟΛΟΓΙΑΣ & ΕΡΕΥΝΑΣ-ΠΑΝΕΠΙΣΤΗΜΙΑΚΕΣ ΕΚΔΟΣΕΙΣ ΚΡΗΤΗΣ, 1η/2018, ΗΡΑΚΛΕΙΟ ΚΡΗΤΗΣ, 77107082
• ΔΙΑΝΥΣΜΑΤΙΚΗ ΑΝΑΛΥΣΗ, Ιωάννης Γιαννούλης, “Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα – Αποθετήριο “”Κάλλιπος”””, Ηλεκτρονικό Βιβλίο/1η έκδοση2016, 320085

 Mandatory written exams at the end of the semester.

http://eclass.uth.gr/eclass/courses/DIB124/