By Lalao Rakotomanana
Across the centuries, the improvement and progress of mathematical ideas were strongly inspired through the wishes of mechanics. Vector algebra used to be built to explain the equilibrium of strength platforms and originated from Stevin's experiments (1548-1620). Vector research was once then brought to check speed fields and strength fields. Classical dynamics required the differential calculus constructed via Newton (1687). however, the concept that of particle acceleration used to be the place to begin for introducing a established spacetime. on the spot speed concerned the set of particle positions in area. Vector algebra thought used to be no longer adequate to check the several velocities of a particle during time. there has been a necessity to (parallel) shipping those velocities at a unmarried aspect ahead of any vector algebraic operation. the suitable mathematical constitution for this delivery was once the relationship. I The Euclidean connection derived from the metric tensor of the referential physique was once the one connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime ideas have been made by means of Einstein in 1905 (special relativity) and 1915 (general relativity) through the use of Riemannian connection. just a little later, nonrelativistic spacetime including the most beneficial properties of common relativity I It took approximately one and a part centuries for connection idea to be authorised as an self sufficient idea in arithmetic. significant steps for the relationship thought are attributed to a sequence of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.
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Extra info for A Geometric Approach to Thermomechanics of Dissipating Continua
The present work rather leans on the assumption of the existence of a boundary mechanical action that is modeled by a 2-form field We on the boundary aB and a boundary heat action also modeled by a 2-form field W H on aB. In the present monograph, the formulation of the cut principle of Euler and Cauchy starts by considering, within the shape of any body B at any given time, a smooth and closed surface. The action of the part outside that surface is equivalent to that of a field of 2-forms (one mechanical, one thermal) defined on the surface.
22) 2. If the derivative of w with respect to the continuum vanishes, then for any p-plet of vectors (u\ , . . , up) embedded in B, the following relation holds: dB -w(u\ , . . up) , dt d = -[w(u\, .. 67), * drpt-to (dB) d * Trw = dt (drpt - tow) = 0 "It :::: to. 25) "It :::: to . 2 Global formulation of conservation laws First, let us recall the integral invariance of a physical quantity. Let v be the velocity field on B engendered by the local group rp. The p-forrn field is an integral invariant by v if it is an integral invariant for every element of the group rp = rpt-to for any t .
Dt For any p-plet (UIO, . , upo) in Bo not depending on t, we then obtain by using the dual definition: drp * (dB -w) (UIO,"" upo) dt d *W)(UIO , ... , upo) . = -(drp dt Remark. 67) allows us to calculate the time derivative with respect to the continuum of a p-form, as the metric tensor, and the volume form by means of the total derivative and via the transposition. 2 Let A be a q-contravariant tensor field on the initial configuration Bo. If the image of A in the actual configuration B, denoted drpA, is applied on 1forms embedded in B, then the derivative of drpA with respect to B is equal to the image of the total derivative of A : dB -(drpA) dt = drp (d- dt A) .
A Geometric Approach to Thermomechanics of Dissipating Continua by Lalao Rakotomanana