ADVANCED MATHEMATICAL GEOLOGY: APPLICATION OF PARTIAL DIFFERENTIAL EQUATION (PDE) AND INTEGRAL CALCULUS TO ROCK FORMING MINERALS WITH RESPECT TO THERMODYNAMIC PRINCIPLE

Chemical reaction in Magma is very complex, because of several anionic and cationic substitutions of different size and charge in the magma and in this case, complex problems require complex solutions. This research was focused on bridging the gap between Bowen and Goldschmidt concepts concerning the problem of elemental substitution and distribution of chemical elements in rocks throughout the time of crystallization with mathematical foundation under thermodynamic change such as Calculus, comprises differentiation, Integration, Cauchy-Riemann equation, Laplace equation and Power series comprises Euler series and Taylor series were applied to predict major minerals encompass the olivine, pyroxene, amphibole, mica, and feldspar, and their associated rocks encompass granite, basalt, andesite, and trachyte and were applied to resolve the problems of rock forming minerals in magma. Findings have shown that, in mathematical context, Bowen’s and Goldschmidt rules were mathematically connected using the Mathematical formulae as; δzδx=limδx→0fx,y+dx,y-f(x,y)δx , and δzδy=limδy→0fx,y+dy-f(x,y)δy The reaction processes using Bowen’s and Goldschmidt rules were called Polymerization and Isomorphous. Since all silicate minerals fall within the space of Pascal triangle, the formation of minerals from melt under thermodynamic condition obeys Cauchy’s definition of sequence throughout the time of crystallization and therefore all mineral have limiting values and are convergent at certain point from 𝛈 = n ⟶ p (ℰz⟶𝛈z) in a complex plane of magma. Each of these magma domains of convergence represents a magma fraction from which crystal nucleation takes place and the resultant crystallization is called fractionation and the sequences of the repeated fractions are called fractional crystallization. Therefore, Cauchy-Riemann conditions must be satisfied at each point Z-Z0 in each domain of magma 𝖂, such that Z, represents a point in which x and y are variable in Z- plane of Magma, 𝖁. Z is called a function of two variables x and y if Z has one definite value for every pair of values of values of x and y as Z fx,y. The variables x and y are called independednt variables, which Z, is called the dependent variable. Finally, the mathematical process, in which minerals Z , crystallized from the magma, 𝖁 and melted back into the homogeneous magma 𝖁 when heated is called CALCULUS in petrology and mineralogy that is differentiation and integration in magma 𝖁. Finally the “mathematical connection” between Bowen’s and Goldschmidt concepts was used in this research for a concise explanation of rock forming minerals from the beginning to the end of crystallization and would help the ‘beginners’ especially students of Earth sciences such as Geology, Mineralogy, Petrology and other chemical sciences such as Chemistry, Geochemistry and Petroleum Geology to have clear understanding of rock forming minerals from the Resident magma (𝕽).