the scalar form of Gauss formula
∰_Ω〖(∂P/∂x+∂Q/∂y+∂R/∂z)dv〗=∯_Σ〖Pdydz+Qdzdx+Rdxdy〗=∯_Σ〖(Pcosα+Qcosβ+Rcosγ)dS〗
After checking the history information, the original prototype of Gauss's theorem was proposed by Lagrange in the seventeenth century. When studying fluids, there was the idea of converting between volume fraction and area fraction. Then Gauss came up with a special form of the theorem in 1813, and finally, in 1826, Michael Ostrogradsky of Russia gave the scalar general form of the divergence theorem in Cartesian coordinates. But they did not have the concept of divergence when they arrived at this theorem. Divergence is a concept in a vector. It is a word created by Maxwell in studying electromagnetic fields. It was originally called convergence and then divergence.
Therefore, from a mathematical point of view, why invent a conversion theorem between triple integral and double integral？ Is it to simplify the triple integral operation？
From a scalar point of view, I don’t understand why three multivariate functions"P(x,y,z)、Q（x,y,z）、R(x,y,z)" should be introduced into the formula.
In addition,why construct such a form of "∂P/∂x+∂Q/∂y+∂R/∂z"in the formula? How did this form come out?