Tensor Calculus Video 89 Notes: Vectors Decomposed into Projections

Overview

• Focus on the decomposition of vectors into projections.
• Examine differences between curves and surfaces.
• Reference to previous concepts from Video 68.

Covariant Derivative

Definition in Plane (Video 68)

• Covariant derivative defined for a simple flat 2D Euclidean manifold.
• Based on the Christoffel symbol: a set of scalar functions used with basis vectors.
• Allows representation of partial derivatives.

Curved Surfaces

• Initial assumption: Christoffel symbol also applicable to curved surfaces.
• Discoveries:
  • The partial derivative may result in a vector pointing in any direction in space.
  • Linear combinations limited to vectors in the tangent plane.
• Modified definition of the Christoffel symbol:
  • Represents the projection of the partial derivative onto the tangent plane.
• New term added for the curvature tensor, resulting in an expression that combines surface projection and normal projection.

Analysis of Curves

Covariant Derivative in Curved Coordinates

• Original definition remains: considers tangent to the curve.
• Need for additional terms to cover other possible vector components:
  • General expression must account for vectors pointing in any direction.
  • Projection onto the curve's tangent line is one-dimensional.

Projection Methodology

• Unlike surfaces, curves do not have a tangent plane or a well-defined normal.
• New approach:
  • Construct a perpendicular plane to the curve at the point of interest.
  • The basis vector acts as the normal of this plane.
  • Define the vector curvature normal by dropping a perpendicular to this new plane.
• Comparison with surface methodology:
  • Surface projections cover two degrees of freedom; curve projections cover one degree of freedom.

Vector Curvature Normal

Orthogonality and Symmetry

• Defined via dot product with contravariant basis vector.
• Results in orthogonality: vector curvature normal is orthogonal to the contravariant basis vector along the curve.
• Notation retains indices for consistency, demonstrating a symmetric relationship but not invariant.

Covariant Derivative of Covariant Basis Vector

Calculation Steps

1. Start with the partial derivative of the covariant basis vector.
2. Incorporate Christoffel symbol definition.
3. Result: Non-metrical nature of the covariant basis vector.
4. Provides explicit relationship for vector curvature normal.

Evaluating Vector Curvature Normal

Final Expressions

• Expressed using Laplacian of position vector.
• Relationship to curvature of a line revisited (Video 76).
• Curvature expressed with principal normal vector, analogous to surface equations.

Summary of Relationships

• Covariant basis vector is not metronilic.
• Vector curvature normal relates to Laplacian of position vector.
• Relationships for curves parallel to those for surfaces, replacing n-hat with p-hat to maintain consistency.
• Preparing to explore surface equations and their applicability to curves in upcoming videos.