Tensor Calculus Video Lectures
A complete 100-lecture series covering tensor calculus from foundations to Riemannian geometry
About This Series
This comprehensive lecture series provides the complete mathematical foundation of tensor calculus required for General Relativity. Starting from basic coordinate systems and building up through covariant derivatives, Christoffel symbols, curvature tensors, and geodesics, these lectures cover everything you need before diving into Einstein's theory.
Recommended Path: Watch Sections I-IV before starting Part I of the GR course. Sections V-VIII align with Parts I-II (Differential Geometry & Curvature).
I. Foundations & Coordinates
Videos 1-7: Why tensors, Euclidean space, tensor fields, and coordinate systems
Video 01: Why Tensor Calculus?
Video Lecture
Why Tensor Calculus
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Video 02: Euclidean Space
Video Lecture
Euclidean Space
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Video 03: Tensor Fields
Video Lecture
Tensor Fields
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Video 04: Affine Coordinates
Video Lecture
Affine Coordinates
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Video 05: Curvilinear Coordinates
Video Lecture
Curvilinear Coordinates
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Video 06: Coordinate Transformations
Video Lecture
Coordinate Transformations
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Video 07: Transformation Examples
Video Lecture
Transformation Examples
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II. Basis Vectors & Transformations
Videos 8-12: Covariant basis, vector transformations, and Jacobians
Video 08: Vector Differentiation
Video Lecture
Vector Differentiation
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Video 09: Covariant Basis
Video Lecture
Covariant Basis
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Video 10: Covariant Basis Examples
Video Lecture
Covariant Basis Examples
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Video 11: Vector Transformations
Video Lecture
Vector Transformations
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Video 12: Jacobian Examples
Video Lecture
Jacobian Examples
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III. Metric Tensor & Tensor Algebra
Videos 13-26: Kronecker delta, dot product, metric tensor, tensor definition, contraction, raising & lowering indices
Video 13: Kronecker Delta
Video Lecture
Kronecker Delta
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Video 14: Dot Product
Video Lecture
Dot Product
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Video 15: Covariant Metric Tensor
Video Lecture
Covariant Metric Tensor
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Video 16: Covariant Metric Examples
Video Lecture
Covariant Metric Examples
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Video 17: Contravariant Basis
Video Lecture
Contravariant Basis
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Video 18: Contravariant Basis Examples
Video Lecture
Contravariant Basis Examples
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Video 19: Tensor Definition
Video Lecture
Tensor Definition
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Video 20: Tensor Algebra
Video Lecture
Tensor Algebra
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Video 21: Contraction
Video Lecture
Contraction
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Video 22: Raising & Lowering Indices
Video Lecture
Raising and Lowering Indices
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Video 23: Tensor Equations
Video Lecture
Tensor Equations
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Video 24: Dot Product (Tensor Formulation)
Video Lecture
Dot Product - Tensor Formulation
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Video 25: Dot Product Examples
Video Lecture
Dot Product Examples
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Video 26: Vector Operation Examples
Video Lecture
Vector Operation Examples
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IV. Covariant Derivatives & Christoffel Symbols
Videos 27-42: Partial derivatives, covariant derivatives, Christoffel symbols, metrinilic property, gradient, trajectories
Video 27: Transformation of Partial Derivatives
Video Lecture
Transformation of Partial Derivatives
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Video 28: Covariant Derivative for Contravariant Vector
Video Lecture
Covariant Derivative for Contravariant Vector
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Video 29: Christoffel Symbol Transformation
Video Lecture
Christoffel Symbol Transformation
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Video 30: Christoffel Symbol Evaluation
Video Lecture
Christoffel Symbol Evaluation
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Video 31: Christoffel Symbol Examples
Video Lecture
Christoffel Symbol Examples
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Video 32: Covariant Derivative for Covariant Vector
Video Lecture
Covariant Derivative for Covariant Vector
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Video 33: Covariant Derivative Proof
Video Lecture
Covariant Derivative Proof
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Video 34: Covariant Derivative Product Rule
Video Lecture
Covariant Derivative Product Rule
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Video 35: Covariant Derivative for Higher Rank Tensors
Video Lecture
Covariant Derivative for Higher Rank Tensors
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Video 36: Covariant Derivative Properties
Video Lecture
Covariant Derivative Properties
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Video 37: Metrinilic Property - Part 1
Video Lecture
Metrinilic Property Part 1
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Video 38: Metrinilic Property - Part 2
Video Lecture
Metrinilic Property Part 2
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Video 39: Gradient
Video Lecture
Gradient
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Video 40: Gradient Examples
Video Lecture
Gradient Examples
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Video 41: Trajectories
Video Lecture
Trajectories
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Video 42: Trajectory Examples
Video Lecture
Trajectory Examples
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V. Determinants, Levi-Civita & Integration
Videos 43-56: Symmetry, determinants, relative tensors, volume element, Levi-Civita symbols, cross product, curl, integration
Video 43: Tensor Symmetry
Video Lecture
Tensor Symmetry
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Video 44: Determinants
Video Lecture
Determinants
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Video 45: Delta System
Video Lecture
Delta System
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Video 46: Relative Tensors
Video Lecture
Relative Tensors
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Video 47: Cofactors
Video Lecture
Cofactors
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Video 48: Volume Element
Video Lecture
Volume Element
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Video 49: Voss-Weyl Formula
Video Lecture
Voss-Weyl Formula
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Video 50: Voss-Weyl Examples
Video Lecture
Voss-Weyl Examples
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Video 51: Levi-Civita Symbols
Video Lecture
Levi-Civita Symbols
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Video 52: Cross Product
Video Lecture
Cross Product
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Video 53: Curl
Video Lecture
Curl
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Video 54: Integration
Video Lecture
Integration
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Video 55: Generalization to Other Dimensions
Video Lecture
Generalization to Other Dimensions
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Video 56: Cross Product & Curl Examples
Video Lecture
Cross Product and Curl Examples
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VI. Embedded Manifolds & Surfaces
Videos 57-73: Tangent space, shift tensor, normals, projection operators, surface Christoffel symbols, surface covariant derivatives
Video 57: Embedded Manifolds
Video Lecture
Embedded Manifolds
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Video 58: Tangent Space
Video Lecture
Tangent Space
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Video 59: Common Baseline
Video Lecture
Common Baseline
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Video 60: Shift Tensor
Video Lecture
Shift Tensor
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Video 61: Sample Metric Objects
Video Lecture
Sample Metric Objects
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Video 62: Embedded Curve
Video Lecture
Embedded Curve
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Video 63: The Normal
Video Lecture
The Normal
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Video 64: Normal Examples
Video Lecture
Normal Examples
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Video 65: Tangent Space Transformations
Video Lecture
Tangent Space Transformations
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Video 66: Projection Operators
Video Lecture
Projection Operators
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Video 67: Projection Operator Examples
Video Lecture
Projection Operator Examples
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Video 68: Christoffel Symbol for Surfaces - Part 1
Video Lecture
Christoffel Symbol for Surfaces Part 1
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Video 69: Christoffel Symbol for Surfaces - Part 2
Video Lecture
Christoffel Symbol for Surfaces Part 2
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Video 70: Surface Christoffel Examples
Video Lecture
Surface Christoffel Examples
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Video 71: Surface Covariant Derivative for Tangent Vectors
Video Lecture
Surface Covariant Derivative for Tangent Vectors
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Video 72: Surface Covariant Derivative for Non-Tangent Vectors
Video Lecture
Surface Covariant Derivative for Non-Tangent Vectors
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Video 73: Surface Covariant Derivative Properties
Video Lecture
Surface Covariant Derivative Properties
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VII. Curvature
Videos 74-87: Weingarten's formula, Gaussian curvature, Riemann-Christoffel tensor, Theorema Egregium, Ricci & Einstein tensors
Video 74: Weingarten's Formula
Video Lecture
Weingarten's Formula
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Video 75: Sample Curvature Tensors
Video Lecture
Sample Curvature Tensors
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Video 76: Curvature Basics
Video Lecture
Curvature Basics
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Video 77: Curvature Using Coordinate Systems
Video Lecture
Curvature Using Coordinate Systems
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Video 78: Gaussian Curvature
Video Lecture
Gaussian Curvature
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Video 79: Curvature Tensor - Part 1
Video Lecture
Curvature Tensor Part 1
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Video 80: Curvature Tensor - Part 2
Video Lecture
Curvature Tensor Part 2
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Video 81: Intrinsic Curvature
Video Lecture
Intrinsic Curvature
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Video 82: Riemann-Christoffel Tensor
Video Lecture
Riemann-Christoffel Tensor
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Video 83: Riemann Normal Coordinates
Video Lecture
Riemann Normal Coordinates
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Video 84: Riemann Tensor Symmetries & Identities
Video Lecture
Riemann Tensor Symmetries and Identities
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Video 85: Commutation for Higher Rank Tensors
Video Lecture
Commutation for Higher Rank Tensors
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Video 86: Theorema Egregium
Video Lecture
Theorema Egregium
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Video 87: Ricci & Einstein Tensors
Video Lecture
Ricci and Einstein Tensors
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VIII. Curves, Geodesics & Riemannian Geometry
Videos 88-100: Embedded curves, Frenet formulas, geodesic curvature, geodesic curves, parallel transport, holonomy, Riemannian geometry
Video 88: Embedded Curves - Part 1
Video Lecture
Embedded Curves Part 1
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Video 89: Curve Projections
Video Lecture
Curve Projections
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Video 90: Embedded Curves - Part 2
Video Lecture
Embedded Curves Part 2
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Video 91: Frenet Formulas
Video Lecture
Frenet Formulas
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Video 92: Curves Embedded in Surfaces
Video Lecture
Curves Embedded in Surfaces
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Video 93: Geodesic Curvature Tensor
Video Lecture
Geodesic Curvature Tensor
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Video 94: Sample Embedded Curve
Video Lecture
Sample Embedded Curve
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Video 95: Geodesic Curves
Video Lecture
Geodesic Curves
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Video 96: Parallel Transport
Video Lecture
Parallel Transport
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Video 97: Parallel Transport Example
Video Lecture
Parallel Transport Example
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Video 98: Holonomy - Part 1
Video Lecture
Holonomy Part 1
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Video 99: Holonomy - Part 2
Video Lecture
Holonomy Part 2
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Video 100: Riemannian Geometry
Video Lecture
Riemannian Geometry
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Mapping to GR Course
Sections I-IV (Videos 1-42)
Watch before starting the GR course. Covers coordinates, basis vectors, metric tensor, covariant derivatives, and Christoffel symbols.
Section V (Videos 43-56)
Aligns with Part I: Differential Geometry. Determinants, volume elements, and Levi-Civita symbols used throughout GR.
Section VI (Videos 57-73)
Aligns with Part I: Manifolds, tangent spaces, and surface geometry are foundational to GR.
Section VII (Videos 74-87)
Aligns with Part II: Curvature of Spacetime. Riemann tensor, Ricci tensor, and Einstein tensor are the heart of GR.
Section VIII (Videos 88-100)
Geodesics, parallel transport, and holonomy are essential for understanding particle motion and curvature effects in GR.