Curvature

Tensor Calculus Lecture 8: Embedded Surfaces and the Curvature Tensor

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• published: 22 Apr 2014
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Textbook: http://bit.ly/ITCYTNew Solutions: http://bit.ly/ITACMS_Sol_Set_YT Errata: http://bit.ly/ITAErrata McConnell's classic: http://bit.ly/MCTensors Weyl's masterpiece: http://bit.ly/SpaceTimeMatter Levi-Civita's classic: http://bit.ly/LCTensors Linear Algebra Videos: http://bit.ly/LAonYT Table of Contents of http://bit.ly/ITCYTNew 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