By Lalao Rakotomanana

Across the centuries, the improvement and progress of mathematical recommendations were strongly influenced via the desires of mechanics. Vector algebra was once constructed to explain the equilibrium of strength platforms and originated from Stevin's experiments (1548-1620). Vector research was once then brought to review pace fields and strength fields. Classical dynamics required the differential calculus constructed through Newton (1687). however, the concept that of particle acceleration was once the start line for introducing a dependent spacetime. immediate pace concerned the set of particle positions in house. Vector algebra conception was once now not adequate to match different velocities of a particle during time. there has been a necessity to (parallel) delivery those velocities at a unmarried element prior to any vector algebraic operation. the suitable mathematical constitution for this shipping used to be the relationship. I The Euclidean connection derived from the metric tensor of the referential physique used to be the one connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime options have been made by way of Einstein in 1905 (special relativity) and 1915 (general relativity) by utilizing Riemannian connection. a little later, nonrelativistic spacetime along with the most gains of normal relativity I It took approximately one and a part centuries for connection conception to be authorized as an self sustaining idea in arithmetic. significant steps for the relationship proposal 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**

**Example text**

In other words, the problem of unicity should not be overlooked. Gurtin and Williams have studied the problem in depth and proposed a theory that admitted continuity assumptions such as the existence of a positive number s. This number s is such that for any separated parts BI and B2 of the whole body B, the intensity of the boundary actions of BI to B2 is lower or equal to the area of contact surface between them multiplied by s. Thus, they have rigorously deduced as theorem [75] that, provided global actions (mechanical C and thermal H) from BI to B2 , there are essentially bounded densities.

153]. Such an invariance allows us to write: r pewo = J",(B) r pewo JB dB -(pewo) dt = O. Second, consider the causes of the changes of this physical quantity. Conservation laws for continuum thermomechanics are usually written in an integral form requiring the integration of scalar and vector fields on a manifold. , [29]. 27) where reWO is the volume source (3-form) and ho(Je) the flux at the boundary (scalaror vector-valued 2-form). 28) as asserting that the rate of increase of the quantity pe in the body B (and more generally in any subbody) may be expressed as the sum of two effects: inflow through the boundary B and growth inside the a 50 3.

Upo) in Bo: drp*W(UIO, . , upo) = w[drp(ulO) , ... ,drp(upo)]. The derivative of the p-form w on B with respect to B is then written as: dB diw(drpulO, ... , drpupo) d dt [w (drpulO , . . , drpupo)] -d [drp *W(UIO, ... , upo)]. 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.