Module Title: Mathematical Analysis II
Type of Module:
X
PC (Prescribed Core Module)
PS (Prescribed Stream Module)
ES (Elective Stream Module)
E (Elective Module)
Level of Module
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Year of Study
2nd
Semester
3rd
Number of credits allocated
6
Name of lecturer / lecturers : Aristophanes Dimakis
--------------------------------------------------------------------------------------------------------------------------------------Description:
An introduction to the notions and techniques of Multivariable Calculus
Prerequisites:
Mathematical Analysis I
Module Contents (Syllabus):
1. Vectors and Motion in the Plane and Polar Functions
Vector-Valued Functions
Modeling Projectile Motion
Polar Coordinates and Graphs
Calculus of Polar Curves
2. Vectors and Motion in Space
Quadric Surfaces in Space
Vector-Valued Functions and Space Curves
Arc Length and the Unit Tangent Vector T
3. Multivariable Functions and Their Derivatives
Functions of Several Variables
Limits and Continuity in Higher Dimensions
Partial Derivatives
The Chain Rule
Directional Derivatives, Gradient Vectors, and Tangent Planes
Linearization and Differentials
Extreme Values and Saddle Points
Lagrange Multipliers
4. Multiple Integrals
Double Integrals
Areas, Moments, and Centers of Mass
Double Integrals in Polar Form
Triple Integrals in Rectangular Coordinates
Masses and Moments in Three Dimensions
Triple Integrals in Cylindrical and Spherical Coordinates
Substitutions in Multiple Integrals
5. Integration in Vector Fields.
Line Integrals
Vector Fields, Work, Circulation, and Flux
Path Independence, Potential Functions, and Conservative Fields
Green's Theorem in the Plane
Recommended Reading:
Α) Principal Reference:
Finney R. L., Weir M. D., Giordano F. R., Thomas' Calculus
Β) Additional References:
Marsden J. E., Tromba A. J., Vector Calculus
Teaching Methods:
Lectures and tutorials
Assessment Methods:
Final exam
Language of Instruction:
Greek
Module Objective (preferably expressed in terms of learning outcomes and competences):
After completion of the course the students will be able to understand notions and apply techniques
related to the following subjects:
1. Vectors and Motion in the Plane and Polar Functions
Vector-Valued Functions
Modeling Projectile Motion
Polar Coordinates and Graphs
Calculus of Polar Curves
2. Vectors and Motion in Space
Quadric Surfaces in Space
Vector-Valued Functions and Space Curves
Arc Length and the Unit Tangent Vector T
3. Multivariable Functions and Their Derivatives
Functions of Several Variables
Limits and Continuity in Higher Dimensions
Partial Derivatives
The Chain Rule
Directional Derivatives, Gradient Vectors, and Tangent Planes
Linearization and Differentials
Extreme Values and Saddle Points
Lagrange Multipliers
4. Multiple Integrals
Double Integrals
Areas, Moments, and Centers of Mass
Double Integrals in Polar Form
Triple Integrals in Rectangular Coordinates
Masses and Moments in Three Dimensions
Triple Integrals in Cylindrical and Spherical Coordinates
Substitutions in Multiple Integrals
5. Integration in Vector Fields.
Line Integrals
Vector Fields, Work, Circulation, and Flux
Path Independence, Potential Functions, and Conservative Fields
Green's Theorem in the Plane