vectors and tensors

vectors and tensors  the dependency chain

that makes vectors and tensors necessary, rather than around historical
chronology. The original sequence runs algebra → calculus → vectors →
quaternions → Maxwell → vector analysis → spacetime → curvature →
tensors → relativity.

TOC

Part I — Representation Debt

  1. Scalar description
    1.1 Magnitude-only quantities
    1.2 Why one number is insufficient
    1.3 Physical versus informational dimensions
    1.4 SOURCE ≠ coordinate description

  2. Coordinates and components
    2.1 Coordinate systems
    2.2 Basis choice
    2.3 Components as representation
    2.4 Object versus component list
    2.5 Representation change as the first GRM transport problem

Part II — Vector Necessity

  1. Vector space
    3.1 Addition and scalar multiplication
    3.2 Linear independence
    3.3 Dimension
    3.4 Basis and coordinate expansion
    3.5 Directed physical quantities as one application, not the definition

  2. Change of basis
    4.1 Same vector, different components
    4.2 Transformation matrices
    4.3 Equivalence across representations
    4.4 Invariant consequences

  3. Dual space
    5.1 Linear functionals
    5.2 Dual basis
    5.3 V versus V*
    5.4 Natural pairing
    5.5 Why vectors and covectors must remain distinct

Part III — Arity Forces Tensors

  1. From one slot to many
    6.1 Linear maps
    6.2 Bilinear maps
    6.3 Multilinear maps
    6.4 Native arity as structural necessity

  2. Tensor product
    7.1 Construction from V and V*
    7.2 Tensor type
    7.3 Covariant, contravariant, mixed
    7.4 Tensor as intrinsic object
    7.5 Components as carrier only

  3. Contraction
    8.1 Natural V*–V pairing
    8.2 Rank reduction
    8.3 Scalar invariants
    8.4 Why contraction does not require a metric

Part IV — Extra Geometry

  1. Metric
    9.1 Bilinear form
    9.2 Length and angle
    9.3 Metric as rank-2 tensor
    9.4 V ↔ V*
    9.5 Raising and lowering
    9.6 Metric is additional structure, not tensor prerequisite

  2. Transformation and invariance
    10.1 Tensor transformation law
    10.2 Same tensor → different basis → different components
    10.3 Same multilinear consequences
    10.4 Symmetry and antisymmetry
    10.5 Coordinate independence

The book itself eventually reaches this modern distinction: tensors are not merely component arrays; their components transform while the intrinsic object is preserved.

Part V — Differential Structure

  1. Tensor fields

  2. Connections

  3. Parallel transport

  4. Covariant derivative

  5. Torsion

  6. Curvature

  7. Metric compatibility

  8. Levi-Civita reconstruction

Part VI — Physical Closure

  1. Stress as bilinear structure

  2. Maxwell: vectors → spacetime tensor

  3. Minkowski metric

  4. Riemann curvature

  5. Ricci contraction

  6. Einstein tensor

  7. Matter-energy tensor

  8. Field equations

  9. Bianchi identities and conservation

  10. Noether symmetry → conserved structure

Final GRM kernel:

object
→ representation
→ transformation
→ invariant relation
→ multilinear arity
→ tensor
→ optional metric
→ connection
→ curvature
→ intrinsic field law.

Quaternions, Maxwell, Minkowski, Ricci, and Einstein should become case studies attached to the dependency they expose, not the spine itself.

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