By Albert Tarantola

While traditional shows of actual theories emphasize the thought of actual volume, this e-book exhibits that there's a lot to realize while introducing the concept of actual caliber. the standard actual amounts easily look as coordinates over the manifolds representing the actual traits. this enables to advance actual theories that experience a level of invariance a lot deeper than the standard one. it really is proven that thoroughly constructed actual theories include logarithms and exponentials of tensors: their conspicuous absence in traditional theories indicates, actually, that the basic invariance precept acknowledged during this booklet is missing in present-day mathematical physics. The booklet reports and extends the speculation if Lie teams, develops differential geometry, providing compact definitions of torsion and of curvature, and adapts the standard proposal of linear tangent program to the intrinsic viewpoint proposed for physics. for example, uncomplicated theories are studied with a few aspect, the speculation of warmth conduction and the idea of linear elastic media. The equations discovered fluctuate quantitatively and qualitatively from these frequently presented.

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**Example text**

This is a (multiplicative) group with, as group operation, the matrix product R2 R1 . It is well known that this group is isomorphic to the group of geometrical 3D rotations. A geometrical rotation is then represented by an orthogonal matrix, and the composition of rotations is represented by the product of orthogonal matrices. 15 for details). This is an o-additive group, with group operation r2 ⊕ r1 = log(exp r2 exp r1 ) . This group is also isomorphic to the group of geometrical 3D rotations.

113), this is the well known “first Bianchi identity” of a manifold. 16 Second Bianchi identity. At any point of a differentiable manifold, the Riemann and the torsion are linked through (jk ) ∇ j Ri mk = ( jk ) Ri mjs Ts k . 115) Contrary to what happens with the first identity, no simplification occurs when using the anassociativity instead of the Riemann. 4 Lie Group Manifolds The elements of a Lie group can be interpreted as the points of a manifold. Lie group manifolds have a nontrivial geometry; they are metric spaces with a curvature so strong that whole regions of the manifold may not be joined using geodesic lines.

114) (the common value being the Jacobi tensor Ji jk ). This is an important identity. 113), this is the well known “first Bianchi identity” of a manifold. 16 Second Bianchi identity. At any point of a differentiable manifold, the Riemann and the torsion are linked through (jk ) ∇ j Ri mk = ( jk ) Ri mjs Ts k . 115) Contrary to what happens with the first identity, no simplification occurs when using the anassociativity instead of the Riemann. 4 Lie Group Manifolds The elements of a Lie group can be interpreted as the points of a manifold.