divergence

divergence

1. Definition

For a vector field $F = P i + Q j + R k$ (components $P,Q,R$ scalar fields with first partials):

is a scalar field — the dot product of $∇$ with $F$. In 2D, $div F = P_x + Q_y$.

2. Physical and Geometric Significance

• Source density: $div F(P)$ = net outward flux per unit volume at $P$, i.e. limit of $∯_{S} F·n dS / Vol$ as volume $→0$. Positive ⇒ source (expansion, fluid created), negative ⇒ sink (compression, fluid lost), zero ⇒ incompressible / solenoidal (no net creation).
• Fluid: If $F$ = velocity field $V$, then $∇·V =0$ ⇔ fluid is incompressible (density constant along flow).
• Electromagnetics: Gauss's law $∇·E = ρ/ε_0$ (electric flux divergence = charge density); $∇·B =0$ (no magnetic monopoles).

3. Properties

• Linearity: $∇·(aF+bG)=a∇·F + b∇·G$.
• Product with scalar $φ$:

• For $F = r = x i+y j+z k$:

in 3D (2 in 2D); for $F = r/r³$:

for $r≠0$ (inverse-square field solenoidal except at origin — point source).

• $∇·(∇×F)=0$ always (divergence of curl zero, uses equality of mixed partials).

4. Worked Examples

(a) F= x² i + y² j + z² k: div F =2x+2y+2z ; at (1,2,3): 12 ✓

(b) F= x i + y j + z k = r: div F =1+1+1=3 (source everywhere, uniform expansion) ✓

(c) F= -y i + x j (2D rotation): P=-y, Q=x ⇒ div=0+0=0 ⇒ solenoidal, incompressible rotation ✓

(d) F= (x/r³) i+(y/r³)j+(z/r³)k = r/r³ , r≠0: P=x/r³ etc. Compute ∂P/∂x =1/r³ -3x²/r⁵ ; sum gives 3/r³ -3r²/r⁵=0 ⇒ div=0 (except singularity at origin where source lives) ✓
    This is key for Gauss divergence theorem to give flux 4π through sphere enclosing origin.

(e) Show F = (x² - y²) i +2x y j + z k ; div=2x+2x+1=4x+1 (not solenoidal generally) ✓

(f) If φ = 1/r, then F=∇φ = -r/r³ ⇒ div F =0 for r≠0 (Laplacian of 1/r zero away from origin).

5. Solenoidal Field

A field with $∇·F =0$ everywhere is called solenoidal. Such a field can be written as $F = ∇×A$ for some vector potential $A$ (Helmholtz).

6. Exam Points

• Define $div F = P_x+Q_y+R_z$ and compute for $F= x i+y j+z k, x² i+...$ — 5 marks, very frequent.
• Physical significance as source density / incompressible condition $∇·V=0$ — 3-5 marks, very frequent.
• Show $∇·(r/r³)=0$ for $r≠0$ — 5 marks, frequent.
• Show $div(∇×F)=0$ and $F$ solenoidal ⇒ existence of vector potential — 5 marks.