By Albert Tarantola
Reviews and extends the speculation of Lie teams, develops differential geometry, providing compact definitions of torsion and of curvature, and adapts the standard inspiration of linear tangent program to the intrinsic perspective proposed for physics.
Uses a unifying representation: uncomplicated theories are studied with a few aspect, the idea of warmth conduction and the idea of linear elastic media.
Shows that the ensuing equations derived during this demeanour fluctuate quantitatively and qualitatively from these frequently presented.
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Additional info for Elements for physics : quantities, qualities, and intrinsic theories
Example text
26) where, as usual, the same symbol is used to denote the components {vi } of a vector v and the components {vi } of the associated form. The gi j are easily shown to be the components of the metric G on the basis {ei ⊗ e j } . 26) is then vi = gi j v j . 28) . 4 Universal Metric for Bivariant Tensors Consider an n-dimensional linear space S . 11 The ‘Frobenius norm’ of a tensor t = {ti j } is defined as t F = gi j gk tik t j . 1 Linear Space 17 If no metric is defined over S , the norm of a vector vi is not defined.
3). 3 that on a manifold with constant torsion, the anassociativity tensor is identical to the Riemann tensor of the manifold (this correspondence explaining the factor 1/2 in the definition of A ). 81). 89) allows to write its definition in terms of the torsion or of the anassociativity as Ji jk = ( jk ) Ti js Ts k = ( jk ) Ai jk . 82) stating the total antisymmetry of the Jacobi tensor now becomes Ji jk = -Ji j k = -Ji k j . 73)) (w ⊕ v)i = wi + vi + ei jk w j vk + qi jk w j wk v + ri jk w j vk v + .
3 Autovector Space The structure about to be introduced, the “space of autoparallel vectors”, is the generalization of the usual structure of (linear) vector space to the case where the sum of elements is not necessarily associative and commutative. 19 Autovector space. Let the set S , with elements u, v, w . . , be a linear space with the two usual operations represented as w+v and λ v . 16 – We say that, while {S, ⊕} is an autovector space, {S, + } is its tangent linear space. When considered as elements of {S, ⊕} , the vectors of {S, +} are also called autovectors.