# Multivariable Calculus

## Course Description:

The Euclidean space Rn. Functions between Euclidean spaces, limit and continuity of functions. Differentiation of vector-valued functions of one variable, applications in mechanics and differential geometry, polar, cylindrical and spherical coordinates. Differentiable functions, partial and directional derivative, the concept of differential. Vector fields, gradient-divergence-curl. Fundamental theorems of differentiable functions: differentiability of composite functions, mean value theorem, Taylor’s formula, implicit function theorems, functional dependence. Local and conditional extremes, Lagrange multipliers. Double and triple integrals: definitions, integrability criteria, properties, change of variables, applications. Contour integrals: Contour integral of the first and second kind, contour integrals independent of path, Green’s Theorem. Elements of surface theory. Surface integrals of the first and second kind. Fundamental theorems of vector analysis (Stokes and Gauss Theorems), applications.

• Semester 2
• Teaching hours 3
• Instructors

### Prerequisite Knowledge

Fundamental knowledge of Mathematical Analysis of real functions of one real variable and Linear Algebra.

### Course Units

# Title Description Hours
1 Introduction The Euclidean space Rn. Functions between Euclidean spaces, limit and continuity of functions. 3
2 Vector-Valued functions Differentiation of vector-valued functions of one variable, applications in mechanics and differential geometry, polar, cylindrical and spherical coordinates cylindrical and spherical coordinates. 6
3 Differentiability of functions of several variables. Differentiable functions, partial and directional derivative, the concept of differential. Vector fields, gradient-divergence-curl. Fundamental theorems of differentiable functions: differentiability of composite functions, mean value theorem, Taylor’s formula, implicit function theorems, functional dependence. 6
4 Extremals Local and conditional extremes, Lagrange multipliers 3
5 Double and triple integrals Double and triple integrals: definitions, integrability criteria, properties, change of variables. 6
6 Contour integrals Contour integrals: Contour integral of the first and second kind, contour integrals independent of path, Green’s Theorem. 6
7 Surface integrals Elements of surface theory. Surface integrals of the first and second kind. Fundamental theorems of vector analysis (Stokes and Gauss Theorems), applications. 9

### Learning Objectives

Rigorous Analysis on the differential and integral calculus of real functions of several variables.

### Teaching Methods

 Teaching methods Lectures in class Blackboard use Yes Yes

### Student Assessment

• Final written exam: 70%
• Mid-term exam: 20%
• Assignments (projects, reports): 10%

### Textbooks - Bibliography

1. Καδιανάκης,Ν., Καρανάσιος, Σ. & Φελλούρης, Α. (2015). Λογισμός Συναρτήσεων πολλών μεταβλητών για τις επιστήμες του μηχανικού, Εκδ. Ν. Καδιανάκης-Σ. Καρανάσιος-Α. Φελλούρης https://service.eudoxus.gr/search/#a/id:41955776/0
2. Παντελίδης, Γ.(2001). Ανάλυση Τόμος ΙΙ, Εκδ. Ζήτη, https://service.eudoxus.gr/search/#a/id:10967/0 3 . Ρασσιάς,Θ. (2014). Μαθηματική Ανάλυση ΙΙ, Εκδ.Τσότρα, https://service.eudoxus.gr/search/#a/id:41955064/0

## Lecture Time - Place:

• Wednesday, 11:45 – 14:30,
Rooms:
• Αμφ. 1/2

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