Differential calculus, absolute

The genesis of this discovery called for a series of curious coincidences. The absolute differential calculus was a speciality only of the School of Zurich, and it required an improbable meeting between this School, Albert Einstein and the mathematician Marcel Grossmann, who was also able to think as a pure physicist. 

Gregorio Ricci-Curbastro. Tensorial Analysis and absolute differential calculus. 

Periodically, scientists uncover, in the treasure troves of mathematicians, a theory that allows the simple solution of a hitherto unresolved problem, or at least makes possible its formulation in a conceptual framework that eventually leads to an elegant solution. A typical example of this process is the adoption of tensor calculus by physicists in the early years of the 20th century. In the 1880s and 1890s, two Italian mathematicians, Gregorio Ricci-Curbastro (1853-1925) and Tullio Levi-Civita (1873-1941), spent years patiently elaborating a mathematical theory initially referred to as absolute differential calculus and later known as tensor calculus. This theory attracted virtually no attention outside of mathematical circles until Albert Einstein realized that it was precisely the tool he crucially needed to develop his general theory of relativity. He... 

The relationship between M and Mv is known from the Equation 40 the absolute activity can be calculated from Equations 48 and 50 by differential calculus. The following formula is obtained ... 

This text assumes a solid background in algebra. All of the mathematical operations required are described in Appendix One or are illustrated in worked-out examples. A knowledge of calculus is not required for use of this text. Differential and integral notions are used only where absolutely necessary and are explained where they are used. 

In order to normalize the residuals so that they don t cancel when one s positive and one s negative (and thus helping to avoid statistical bias), we are usually concerned with the square of Rk when doing least-squares regression. We use squared terms and not absolute values so that the function is differentiable, don t worry about this if you haven t taken calculus yet. 

The existence of states that are inaccessible to adiabatic processes was shown by Carath odory to be necessary and sufficient for the existence of an integrating factor that converts into an exact differential [2-4]. From the calculus we know that for differential equations in two independent variables, an integrating factor always exists in fact, an infinite number of integrating factors exist. Experimentally, we find that for pure one-phase substances, only two independent intensive properties are needed to identify a thermod)mamic state. So for the experimental situation we have described, we can write SQ gj, as a function of two variables and choose the integrating factor. The simplest choice is to identify the integrating factor as the positive absolute thermodynamic temperature X = T. Then (2.3.3) becomes... 

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