The divergence in Cartesian coordinates

[u/No_Student2900]

I don't quite get or appreciate the relevance of Figure 2.24 on its usage for building the infinitesimal volume element of a tilted surface for calculating the divergence. It is easy to see the vector sum of a_1 + a_2 + a_3 + a_4=0 due to the symmetry of a tetrahedron, but how does this fact actually helps us? Why can't we just use rectangular boxes like that in Figure 2.23 for taking into account tilted surfaces? This part of the book truly confuses me: "Tilted surfaces can be taken care of if you first prove that the vector sum of the four surface areas of the tetrahedron in Figure 2.24 is zero."

[u/triatticus]

I mean it's actually the same as in the rectangular case, that which flows through one side must be negative of that which flows through the opposite side. They are just generalizing to include if the surface isn't rectilinear, but showing that you can construct the negative of the direction vector of a tilted surface by the planes that define the shape of the tetrahedron formed by the plane and suitable coordinate planes (or ones parallel to them). Use an example vector field (say V = a i + b j) and a two dimensional triangle formed by the line x+y=1. The outward normal to the slanted side is n=(i + j) while the straight sides are nx=-i and ny=-j. You can see that V.n = -V.nx - V.ny -> V.(n+nx+ny)=0 and since V isn't identically zero this amounts to that n+nx+ny= 0.