Microfluidics: Modeling, Mechanics and Mathematics

we can therefore propose the reconstruction function ${\tilde{g}}^{\left(\Delta \right)}$ as a function of the Lagrangian coordinates ${\displaystyle {\xi }_A},{\displaystyle {\xi }_B}$ , and ${\displaystyle {\xi }_C}$ using Eq.1 as

${\tilde{g}}^{\left(\Delta \right)}\left({\displaystyle {\xi }_A},{\displaystyle {\xi }_B},{\displaystyle {\xi }_C}\right)=g^{\left(\Delta ,A\right)}{\displaystyle {\xi }_A}+g^{\left(\Delta ,B\right)}{\displaystyle {\xi }_B}+g^{\left(\Delta ,C\right)}{\displaystyle {\xi }_C}$

  (Eq. 32.52)

where A, B, and C are the vertices of the triangle. We can then calculate the derivatives as

$\frac{\partial \tilde{\mathrm{g}}\left(\mathrm{\Delta }\right)}{\partial x}={\displaystyle g^{\left(\Delta ,A\right)}}\frac{\partial {\xi }_A}{\partial x}+{\displaystyle g^{\left(\Delta ,B\right)}}\frac{\partial {\xi }_B}{\partial x}+{\displaystyle g^{\left(\Delta ,C\right)}}\frac{\partial {\xi }_C}{\partial x}$

$\frac{\partial \tilde{\mathrm{g}}\left(\mathrm{\Delta }\right)}{\partial y}={\displaystyle g^{\left(\Delta ,\mathrm{A}\right)}}\frac{\partial {\xi }_A}{\partial y}+{\displaystyle g^{\left(\Delta ,\mathrm{B}\right)}}\frac{\partial {\xi }_B}{\partial y}+{\displaystyle g^{\left(\Delta ,\mathrm{C}\right)}}\frac{\partial {\xi }_C}{\partial y}$

Using Eq. 32.5 ...