Gauss divergence theorems

1. Statement — Gauss Divergence Theorem (Ostrogradsky)

Let $V$ be a bounded, closed, piecewise smooth solid with outward oriented closed surface $S = ∂V$ . If $F = P i+Q j+R k$ has continuous first partials on an open region containing $V$ , then:

∯_{S} F·n dS = ∭_{V} ∇·F dV = ∭_{V} (∂P/∂x+∂Q/∂y+∂R/∂z) dV

i.e. total outward flux = volume integral of divergence (source density integrated). $∯$ means closed surface.

2. Geometric Meaning

Integrate source density $∇·F$ over volume ⇒ net flux out. If $∇·F>0$ inside, more exits than enters (source); if $=0$ , net flux zero (solenoidal, what goes in comes out).

3. Working Steps (recipe)

Compute $∇·F = P_x+Q_y+R_z$ (often constant/simple).
Describe $V$ (Cartesian/cylindrical/spherical) and evaluate $∭_{V} ∇·F dV$ (often far easier than surface integral).
That value equals $∯_{S}F·n dS$ .

Conversely, to find volume integral of divergence, compute flux — whichever easier.

4. Worked Examples

(a) F= r = x i+y j+z k, V: ball x²+y²+z²≤a²: ∇·F=3 ⇒ ∯_{S}F·n dS =∭_{V}3 dV =3·4πa³/3=4πa³ ✓ matches earlier direct flux a·area.

(b) F= x² i+ y² j+ z² k, V: cube 0≤x,y,z≤1: ∇·F=2x+2y+2z ⇒ ∭=∫_{0}^{1}∫_{0}^{1}∫_{0}^{1}(2x+2y+2z)dz dy dx =3 ✓
    Direct flux through 6 faces sums to same but longer.

(c) F= -y i+ x j ??? Need divergence zero ⇒ flux zero for any closed surface not enclosing singularity? Check.

(d) F= r/r³ , r≠0: ∇·F=0 except at origin ⇒ ∯_{sphere}F·n dS=4π (point source) but ∭_{V}0=0 if origin excluded? Shows need to handle singularity — Gauss fails if singularity inside, need punctured volume.

(e) Electrostatics: F=E= q r/(4π ε₀ r³) ⇒ ∇·E=0 away from charge, but ∯ E·n dS = q/ε₀ via singularity — Gauss's law.

5. Relation to Green (2D)

Flux form of Green in plane is 2D Gauss: $∮_{C} F·n ds = ∬_{D} ∇·F dA$ .

6. Exam Points

State Gauss divergence theorem with outward $n$ — 3-5 marks, almost certain.
Evaluate $∯_{S}F·n dS$ via $∭_{V}∇·F dV$ for sphere/cube/cylinder — 7-10 marks, very frequent (often "using Gauss" compulsory).
Show $∇·(r/r³)=0$ for $r≠0$ but flux =4π — 5 marks, frequent (handling singularity).
Volume via $∯$ or vice versa — 5 marks.

Gauss divergence theorems