User talk:Rschwieb/Basic GA

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Exercises in GA

Some background

For simplicity, Gibbs's vector algebra, geometric algebra will be considered over the same set: scalars, vectors etc. This will simplify reasoning. The only difference is the operations defined. The number of dimensions and the metric will in general be unspecified other than being nondegenerate.

The expansion to bivectors etc. will be assumed, though these are not used in Gibbs's algebra. Axial vectors are sometimes treated as a separate vector space, but here it will be simpler to have them live in 1-vector space as is normal at an elementary level (the sign of the axial vector is handled by the conversion to a bivector).

Defined notation, operations and operators

Conventions (apply unless stated):

• Greek super- and subscripts indicate indexing over the full range of indices.
• The (Einstein's) summation convention applies for a repeated Greek index.
• The (Penrose's) abstract index notation is indicated with Latin super- and subscripts, i.e. h ^b = h ^βe_β , with contraction indicated by a repeated abstract index.
• We will assume here that ∂_μ e ^μ = 0.
• Several indices on e are shorthand for a geometric product, e.g. e_θφ ^ψ = e_θ e_φ e^ψ

Symbols:

• +: addition of any elements, shared by all the algebras
• ·: inner, dot or scalar product on vectors, defined by a bilinear form
• Juxtaposition: geometric product
• ×: cross product of Gibbs's algebra; defined only in three dimensions
• ∧: exterior or wedge product, also called outer product :( in geometric algebra
• e_μ: a basis of the 1-vector space, not in general assumed orthonormal
• e ^μ: a basis of the dual 1-vector space: e ^μe_ν  = δ ^μ_ν 
• ∂_μ  denotes the partial derivative ^∂
  _∂x ^μ
• ∇: the operator e ^μ∂_μ 
• ε_αβγ : the Levi-Civita symbol

Definitions in terms of basis

The geometric product

e_α e_β  = ?

The dot product

e_α  ⋅ e_β  = ⟨e_α , e_β ⟩

The wedge product

e_α  ∧ e_β  = ?

The cross product

e_α  × e_β  = ε_αβγ e^γ 

The ∇ operator can be applied using any of the multiplication operators in a pretty obvious way. For example:

• ∇s = e ^μ∂_μ s
• ∇ · s = e ^μ · ∂_μ s
• ∇ ∧ s = e ^μ ∧ ∂_μ s
• ∇ × s = e ^μ × ∂_μ s

Exercises and questions

Please go through any aspects you want to review or have me describe (visualization), or ask me to set problems. My problem is that I have no idea where to start, since you have more knowledge that I have in pretty much every math area, but you clearly want to gain familiarity with some things so that they are automatic. There are some concepts I may thow in, such as ∇ being defined as generalized directional derivative. — Quondum^t_c 19:33, 17 December 2011 (UTC)

A wacko question: What are the implications of the bilinear form being nonsymmetric?

An exercise: complete a consistent definition of geometric product in this setting, with reference only to the bilinear form, not the other products or a tensor product. Then formulate relationships between all the products.