By Lalao Rakotomanana

Across the centuries, the advance and progress of mathematical ideas were strongly prompted by means of the desires of mechanics. Vector algebra used to be constructed to explain the equilibrium of strength platforms and originated from Stevin's experiments (1548-1620). Vector research was once then brought to review speed fields and strength fields. Classical dynamics required the differential calculus built by means of Newton (1687). however, the concept that of particle acceleration used to be the place to begin for introducing a established spacetime. prompt pace concerned the set of particle positions in house. Vector algebra concept used to be no longer adequate to match different velocities of a particle during time. there has been a necessity to (parallel) delivery those velocities at a unmarried element ahead of any vector algebraic operation. the ideal mathematical constitution for this shipping used to be 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 options have been made via Einstein in 1905 (special relativity) and 1915 (general relativity) by utilizing Riemannian connection. just a little later, nonrelativistic spacetime along with the most good points of basic relativity I It took approximately one and a part centuries for connection thought to be authorised as an self reliant conception in arithmetic. significant steps for the relationship idea are attributed to a chain of findings: Riemann 1854, Christoffel 1869, Ricci 1888, Levi-Civita 1917, WeyJ 1918, Cartan 1923, Eshermann 1950.

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**Additional resources for A Geometric Approach to Thermomechanics of Dissipating Continua**

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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) . 5 Kinematics of continuum Proof Consider the q-plet of I-fonns (wI, . ,wq ) in the actual configuration B . Then let us apply the tensor drpA on these fonns: drpA(w l , ... , w q ) = A(drp*w l , ...

2 (Conservation oflinear momentum) Let B be anypartofa moving continuum. The rate of change of B 's linear momentum equals the resultant of the body and contact forces on B as B moves with the continuum. 3. For angular momentum. By considering now the angular momentum about any fixed point A in the ambient space, we introduce the variables: e = wo(AM, v, ei) = Ii re = wo(AM, pb, ei) Je = AM 1\ Pni. Angular momentum's conservation law holds: { dB (pliwo) dt iB = { p(AM X b)iWO + { iB iB div (AM x Pni )wo.

1. Classes of maps based on the existence and properties of the transfonnation rp and its differential d rp. By addressing the various classes of transfonnations, it is then possible to address the fundamental problem of modeling the irreversibility in continuum mechanics. Given the two configurations, initial Bo and actual B, of the continuum, with a map between them, it is worthwhile to assess if it is possible to uniquely detennine the tensorial rule over the actual configuration in tenns of tensorial rule given over the initial state.