I have to integrate $$ \int_S^\ x~dy~dz + y~dz~dx + z~dx~dy\ $$ where '$S$' is the cube of unit length.
I have the solution of the problem which converts this surface integral into a volumetric integral using the gauss divergence theorem, in this way $$ \nabla\cdot F = 1 + 1 + 1 = 3 $$
$$\int_V\ \nabla \cdot F~dV = \int_V3~dV = 3\cdot 1 = 3 $$
From what I have been doing till now is that I used to have a vector on which I applied the gauss divergence theorem where $\hat{i}$,$\hat{j}$ and $\hat{k}$ were defined.
How did they apply $ \nabla \cdot F $ to a scalar? It makes no sense to me.
Thank You
Best Answer
$\newcommand{\Vec}[1]{\mathbf{#1}}$The integral $$ \iint_{S} x\, dy\, dz + y\, dz\, dx + z\, d\, dy $$ is a vector surface integral, giving the flux of the radial field $F(x, y, z) = x\Vec{i} + y\Vec{j} + z\Vec{k}$ over the surface of the unit cube. This explains the Gauss' theorem calculation you sketch.
Note generally that if $S$ is an oriented surface with continuous unit normal field $\Vec{n}$, and if $F$ is a continuous vector field on $S$, then $$ \iint_{S} F \cdot d\Vec{S} = \iint_{S} (F \cdot \Vec{n})\, dS $$ (once expressed using a parametrization) is the integral of a scalar function, the flux density $F \cdot \Vec{n}$, over $S$.
If you prefer, the terms "scalar line/surface integral" and "vector line/surface integral" refer only to how a particular integral arises. After conversion to a Cartesian integral using a parametrization, each is the integral of a scalar function, and has a numerical value.