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Course11110476545 lectures

Geometry of Curves and Surfaces

Prof. Abhijit Pal · IIT Kanpur

Start with lecture 1 At NPTEL

Lectures

Week 1

  1. Lecture 1 : Parametrized Curves and Differentiable Curves YouTube
  2. Lecture 2 : Length of a Curve YouTube
  3. Lecture 3 : Regular Curve, Arc-length, Re-parametrization of a Curve YouTube
  4. Lecture 4 : Unit Speed Re-parametrization, Curvature of a Curve YouTube

Week 2

  1. Lecture 5 : Curvature of a regular curve & Curvature formulas YouTube
  2. Lecture 6 : Level curves, Implicit function theorem YouTube
  3. Lecture 7 : Orientation of a curve, Signed curvature, Tangent & Normal vectors YouTube
  4. Lecture 8 : Turning Angle YouTube

Week 3

  1. Lecture 9 : Curve determined by curvature, Total Curvature YouTube
  2. Lecture 10 : Hopf's Rotation Index Theorem YouTube
  3. Lecture 11 : Four Vertex Theorem YouTube
  4. Lecture 12 : Isoperimetric Inequality YouTube

Week 4

  1. Lecture 13 : Space Curves & Principal Vectors YouTube
  2. Lecture 14 : Frenet-Serret Equations YouTube
  3. Lecture 15 : Curve determined by curvature and torsion YouTube
  4. Lecture 16 :Several variable calculus YouTube

Week 5

  1. Lecture 17 : Regular surfaces in real three dimensional space YouTube
  2. Lecture 18 : Examples of regular surfaces YouTube
  3. Lecture 19 : Regular Value, Critical Value of a differentiable map YouTube
  4. Lecture 20 : Regular surface locally graph of a differentiable function YouTube

Week 6

  1. Lecture 21 : Change of parametrizations in regular surfaces YouTube
  2. Lecture 22 : Differentiable function on surfaces YouTube
  3. Lecture 23 : Surface of Revolution, Torus, Higher genus surfaces YouTube

Week 7

  1. Lecture 24 : Tangent Plane and Tangent Vectors YouTube
  2. Lecture 25 : Examples of Tangent Planes YouTube
  3. Lecture 26 : Differential Map between Tangent Spaces YouTube
  4. Lecture 27 : Orientation of Surfaces and Normal Vector Field YouTube

Week 8

  1. Lecture 28 : First Fundamental Form on Surfaces YouTube
  2. Lecture 29 : Change of Parametrization in First Fundamental Form and Area YouTube
  3. Lecture 30 : Area of Some Surfaces and Gradient on Surfaces YouTube
  4. Lecture 31 : Isometries, local isometries, conformal maps YouTube

Week 9

  1. Lecture 32 : Gauss Map and its Fundamental Properties YouTube
  2. Lecture 33 : Self Adjoint Operator YouTube
  3. Lecture 34 : Second Fundamental Form and normal Curvature YouTube

Week 10

  1. Lecture 35 : Examples for second fundamental form YouTube
  2. Lecture 36 : Principal curvatures and Principal Directions YouTube
  3. Lecture 37 : Examples for principal curvatures and principal directions YouTube

Week 11

  1. Lecture 38 : Gaussian Curvature and Mean Curvature YouTube
  2. Lecture 39 : Christoffel Symbols and The Gauss's Remarkable Theorem YouTube
  3. Lecture 40 : Invariance of Gaussian Curvature and Examples of Christoffel Symbols YouTube
  4. Lecture 41 : Vector Field and Covariant Derivative YouTube

Week 12

  1. Lecture 42 : Vector Field along a curve, parallel vector field and geodesics YouTube
  2. Lecture 43 : Geodesics and its equations YouTube
  3. Lecture 44 : Simplicial Complex, Triangulation and Euler Characteristics YouTube
  4. Lecture 45 : Gauss Bonnet Theorem YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Kanpur
Department
Mathematics
Teacher
Prof. Abhijit Pal
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
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