(MATH 521) | GSÜ - ECTS Information Guide

Content

Objective

-This course aims to introduce the fundamental concepts of differential geometry and the study of differentiable manifolds. Students will learn to use tangent spaces, vector fields, differential forms, and Stokes’ theorem in geometric problems.

Course Content

-Differentiable manifolds, charts and atlases; smooth maps; submanifolds; tangent spaces and tangent maps; submersions, immersions, and embeddings; tangent bundles and vector fields; integral curves and flows; differential forms, exterior algebra, pullbacks, and exterior differentiation; orientation and integration on manifolds; Stokes’ theorem; an introduction to de Rham cohomology; applications to curves and surfaces.

Course Learning Outcomes

Upon successful completion of this course, students will be able to:

Define differentiable manifolds, charts, and atlases.
Compute tangent spaces and tangent maps.
Use the notions of submanifolds, immersions, submersions, and embeddings.
Work with vector fields and differential forms.
Use exterior differentiation, orientation, and integration on manifolds.
Apply Stokes’ theorem to elementary geometric problems.

Teaching and Learning Methods

Lectures, board-based proofs and examples, guided problem solving, weekly exercises, and in-class discussions. Theoretical concepts are illustrated through examples such as spheres, tori, projective spaces, curves, and surfaces.

Weekly Contents

Theory Topics
Week	Weekly Contents
1	Review of multivariable calculus: differentiability, inverse function theorem, and constant rank theorem.
2	Differentiable manifolds: charts, atlases, changes of coordinates, and fundamental examples.
3	Smooth maps between manifolds; submanifolds and examples defined by equations.
4	Tangent spaces: definitions using curves and derivations.
5	Tangent maps, chain rule, and the inverse function theorem on manifolds.
6	Submersions, immersions, embeddings, and the regular value theorem.
7	Tangent bundles, vector fields, integral curves, and an introduction to flows.
8	Midterm
9	Cotangent spaces, differential forms, and exterior algebra.
10	Pullbacks of forms, exterior derivative, closed forms, and exact forms.
11	Orientation, manifolds with boundary, and integration of differential forms.
12	Stokes’ theorem and classical applications.
13	Introduction to de Rham cohomology and elementary examples.
14	Applications to the geometry of curves and surfaces; general review.

Assessment System

Contribution to Overall Grade
Activities	Number	Contribution
Contribution of in-term studies to overall grade	2	50
Contribution of final exam to overall grade	1	50
Total	3	100

In-Term Studies
Activities	Number	Contribution
Assignments	7	7
Presentation	1	10
Midterm Examinations (including preparation)	1	10
Project	0	0
Laboratory	0	0
Other Applications	0	0
Quiz	0	0
Term Paper/ Project	0	0
Portfolio Study	0	0
Reports	0	0
Learning Diary	0	0
Thesis/ Project	0	0
Seminar	0	0
Other	0	0
Make-up	0	0
Total	9	27

Relation of Proficiency

No	Program Learning Outcomes	Contribution
No	Program Learning Outcomes	1	2	3	4	5
1					X
2				X
3				X
4					X
5					X
6			X
7
8		X
9
10			X
11					X
12
13				X
14				X
15
16		X
17

ECTS

Activities	Number	Period	Total Workload
Class Hours	14	3	42
Working Hours out of Class	10	3	30
Assignments	7	5	35
Presentation	1	10	10
Midterm Examinations (including preparation)	1	12	12
Project	0	0	0
Laboratory	0	0	0
Other Applications	0	0	0
Final Examinations (including preparation)	1	10	10
Quiz	0	0	0
Term Paper/ Project	0	0	0
Portfolio Study	0	0	0
Reports	0	0	0
Learning Diary	0	0	0
Thesis/ Project	0	0	0
Seminar	0	0	0
Other	0	0	0
Make-up	0	0	0
Yıl Sonu	0	0	0
Hazırlık Yıl Sonu	0	0	0
Hazırlık Bütünleme	0	0	0
Total Workload			139
Total Workload / 25			5.56
Credits ECTS			6