By Lalao Rakotomanana
Across the centuries, the advance and progress of mathematical ideas were strongly encouraged via the wishes of mechanics. Vector algebra was once constructed to explain the equilibrium of strength structures and originated from Stevin's experiments (1548-1620). Vector research used to be then brought to review speed fields and strength fields. Classical dynamics required the differential calculus constructed via Newton (1687). however, the idea that of particle acceleration used to be the start line for introducing a dependent spacetime. immediate pace concerned the set of particle positions in area. Vector algebra idea was once now not enough to match different velocities of a particle during time. there has been a necessity to (parallel) shipping those velocities at a unmarried aspect sooner than any vector algebraic operation. the correct mathematical constitution for this shipping was once the relationship. I The Euclidean connection derived from the metric tensor of the referential physique was once the single connection utilized in mechanics for over centuries. Then, significant steps within the evolution of spacetime options have been made by means of Einstein in 1905 (special relativity) and 1915 (general relativity) by utilizing Riemannian connection. a bit of 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 accredited as an self sufficient conception 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 resources for A Geometric Approach to Thermomechanics of Dissipating Continua
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) .
Geometry and Kinematics 1. 27) 2. , , a translational holonomy being defined as a nonclosure of a spacetime loop into a flat spacetime. It should be stressed that describing a continuum with singularity by simply measuring lengths with a metric tensor g is not enough. In usual continuum theory devoted to nondefected (noncracked solids, ideal fluids without vortices), it is common to choose a Euclidean affine connection to calculate derivatives of tensor fields. As soon as singularities are present in the bulk, this situation changes.
28). 28), we have respectively: 1. For mass. By replacing e = 1, re = 0, J e = 0, mass conservation holds: 1 B dB -(pwo) dt = O. 1 (Mass conservation) The mass of any part B of a continuum does not change as B moves with the continuum. 2. For linear momentum. By using the definition of a vector-valued I-form, we can consider the linear momentum (per unit mass) by introducing the variables e = pv i , re = pb i , and J e = p~. The conservation law for linear momentum then holds: 1 B d B. -(pv'wo) dt = l' +1 .