Tensor Calculus Lecture 14f: Principal Curvatures

Rules of the Game
Coordinate Systems and the Role of Tensor Calculus
Change of Coordinates
The Tensor Description of Euclidean Spaces
The Tensor Property
Elements of Linear Algebra in Tensor Notation
Covariant Differentiation
Determinants and the Levi-Civita Symbol
The Tensor Description of Embedded Surfaces
The Covariant Surface Derivative
Curvature
Embedded Curves
Integration and Gauss’s Theorem
The Foundations of the Calculus of Moving Surfaces
Extension to Arbitrary Tensors
Applications of the Calculus of Moving Surfaces

Index:
Absolute tensor
Affine coordinates
Arc length
Beltrami operator
Bianchi identities
Binormal of a curve
Cartesian coordinates
Christoffel symbol
Codazzi equation
Contraction theorem
Contravaraint metric tensor
Contravariant basis
Contravariant components
Contravariant metric tensor
Coordinate basis
Covariant basis
Covariant derivative
Metrinilic property
Covariant metric tensor
Covariant tensor
Curl
Curvature normal
Curvature tensor
Cuvature of a curve
Cylindrical axis
Cylindrical coordinates
Delta systems
Differentiation of vector fields
Directional derivative
Dirichlet boundary condition
Divergence
Divergence theorem
Dummy index
Einstein summation convention
Einstein tensor
Equation of a geodesic
Euclidean space
Extrinsic curvature tensor
First groundform
Fluid film equations
Frenet formulas
Gauss’s theorem
Gauss’s Theorema Egregium
Gauss–Bonnet theorem
Gauss–Codazzi equation
Gaussian curvature
Genus of a closed surface
Geodesic
Gradient
Index juggling
Inner product matrix
Intrinsic derivative
Invariant
Invariant time derivative
Jolt of a particle
Kronecker symbol
Levi-Civita symbol
Mean curvature
Metric tensor
Metrics
Minimal surface
Normal derivative
Normal velocity
Orientation of a coordinate system
Orientation preserving coordinate change
Relative invariant
Relative tensor
Repeated index
Ricci tensor
Riemann space
Riemann–Christoffel tensor
Scalar
Scalar curvature
Second groundform
Shift tensor
Stokes’ theorem
Surface divergence
Surface Laplacian
Surge of a particle
Tangential coordinate velocity
Tensor property
Theorema Egregium
Third groundform
Thomas formula
Time evolution of integrals
Torsion of a curve
Total curvature
Variant
Vector
Parallelism along a curve
Permutation symbol
Polar coordinates
Position vector
Principal curvatures
Principal normal
Quotient theorem
Radius vector
Rayleigh quotient
Rectilinear coordinates
Vector curvature normal
Vector curvature tensor
Velocity of an interface
Volume element
Voss–Weyl formula
Weingarten’s formula

Applications: Differenital Geometry, Relativity

[darrenpeck]

I always enjoy your wonderful lectures. Thank you

[TlnITA]

10:15 Shouldn't k_1 and k_2 be the eigenvalues of B super α sub β, instead of the covariant version ?

[happyhayot]

Watching this 8 year later and realizing there's no differential geometry playlist 🤧. Would've been awesome, I could easily binge all your videos.

[sailingonthoughts]

I'm honestly gonna watch every relevant thing on this channel just to see if there is something I missed. Probably on 1.5 speed tho, the lecturer is extremely clear but talk too slow for my liking, I understand the reason behind it and I think it is probably for the best. Just want to say I really appreciate you for what you do and I wish more people can find you, I will recommand you to any friend or colleague that need help with some of the topics you have.

[marxman]

On page 232 of the text book, eqn 13.120. Clearly explained why the eigenvalues are the answers to the optimization problem.

[sailingonthoughts]

I wish I found this serie like 2 years ago, had to learn some multilinear ml and other stuff for some stuff and had to dig through multiple textbook and wiki pages. Some books are filled with implementation but no insight, some I had to google everything in appendix. It's almost a norm for me but I wish it isn't. A proper course is so important if you want to really understand what you are doing.

[esadcorhodzic]

I'm an engineering student who has thoroughly enjoyed your book and lectures on Tensor Calculus. Thank you for using an intuitive approach that isn't purely proof based as in many other texts. That being said do you know of any texts/resources on topology that use a similar teaching style?