surface and volume integrals

surface and volume integrals

1. Surface Integral — Two Types

For a surface $S$ with unit normal $n$ (choose orientation) and $dS$ area element, or vector $dS = n dS$:

• Scalar (flux of something through surface with weight $φ$):

for parametrised $r(u,v)$, $(u,v)∈D$.

• Vector (flux): for vector field $F$, flux outward/inward is:

gives net outward flux if $n$ outward (e.g., fluid volume rate).

For explicit $z = g(x,y)$ over $D$ in $xy$-plane, take $F(x,y)=(x,y,g(x,y))$, then $dS = √[1+g_x²+g_y²] dx dy$ and $n dS = (-g_x,-g_y,1) dx dy$.

2. Volume Integral

We already met $∭_{V} φ dV$ — volume integral of scalar field $φ$ (mass if $φ$ density). For vector, $∭_{V} F dV$ is vector sum, less common than $∭ ∇·F dV$ (Gauss).

3. Worked Examples

(a) Flux of F= r through sphere S: x²+y²+z²=a² outward: F·n = r·(r/a)= a ⇒ ∬_{S} F·n dS = a·area =a·4πa²=4πa³ ✓
    Via divergence theorem later: ∭_{V} ∇·F dV =∭ 3 dV =3·4πa³/3=4πa³ same.

(b) Surface integral ∬_{S} φ dS where φ= z over hemisphere z=√(a² -x² -y²): D disc x²+y²≤a²
    dS = a/√(a² -r²)?Actually √(1+g_x²+g_y²)= a/√(a² -r²)? Compute: g=√(a² -r²), g_r=-r/√..., so √(1+g_r²)=a/√... ⇒ integral =2π a³/3? Check: ∬ z dS =2π a³/3 ✓

(c) Volume: ∭_{V} 1 dV = Vol(V); cylinder r≤a,0≤z≤h gave πa² h via cylindrical (topic 8).

(d) Surface area itself: ∬_{S} 1 dS = area; for cone z=r, 0≤z≤h, area =π a·l where l=√(a²+h²) ✓

(e) Need to choose n direction: for closed surface, outward is default (Gauss); for open surface, orientation follows right-hand rule with boundary C (Stokes).

4. Relation to Previous

• Line integral moves along curve; surface integral moves across surface (flux). Both become double integrals via parametrisation.
• Divergence and curl theorems (next) convert surface/line to volume/area, saving work when direct parametrisation is messy.

5. Exam Points

• Evaluate $∬_{S} F·n dS$ for $F= x i+y j+z k$ over sphere/cube/paraboloid — 7 marks, very frequent (often doable directly or via Gauss).
• Evaluate $∬_{S} φ dS$ and surface area — 5 marks, frequent.
• Volume $∭_{V} dV$ vs surface — 5 marks, frequent.
• State $dS =|r_u×r_v| du dv$ and for $z=g(x,y)$ $dS=√(1+g_x²+g_y²) dx dy$ — 3-5 marks.