Stokes'

Stokes'

1. Statement — Stokes' Theorem

Let $S$ be an oriented, piecewise smooth open surface bounded by a closed, piecewise smooth curve $C$ with orientation given by right-hand rule (fingers along $C$, thumb along $n$). If $F = P i+Q j+R k$ has continuous first partials on an open region containing $S$, then:

The line integral around $C$ (circulation) equals the flux of $curl F$ through any surface spanning $C$. For $F$ in $xy$-plane, this reduces to Green's theorem.

2. Geometric Meaning

$curl F$ measures local rotation; Stokes says total swirl through $S$ equals circulation around its edge. If $∇×F=0$ on $S$, then $∮_{C}F·dr=0$ for any bounding $C$ (conservative).

3. Working Steps (recipe)

1. Choose $S$ spanning $C$ (often the planar disc bounded by $C$ is easiest).
2. Compute $∇×F$.
3. Find unit normal $n$ (right-hand rule decides sign) and $dS$ (for $z=g(x,y)$, $n dS = (-g_x,-g_y,1)dx dy$).
4. Evaluate $∬_{S} (∇×F)·n dS$.

If $C$ is given as intersection of two surfaces, the spanning surface is either.

4. Worked Examples

(a) F= -y i+ x j, C: circle x²+y²=a² in plane z=0, counter-clockwise when seen from +z.
    ∇×F=2k. Choose S = disc in plane z=0, n=k, (∇×F)·n=2 ⇒ ∬_{S}2 dS=2·πa²=2πa² ✓ matches line integral 2πa² from earlier.

(b) F= y i+ z j+ x k, C: intersection of sphere x²+y²+z²=9 and plane z=0?? Actually C is circle. Choose S = disc? Need compute: ∇×F= (-1+?) Compute: R_y -Q_z =0-1=-1 for i, P_z -R_x=0-1=-1 j, Q_x -P_y=0-1=-1 k ⇒ (-1,-1,-1); flux through disc n=k gives -1·area.

(c) Verify for hemisphere S: x²+y²+z²=a², z≥0 bounded by circle C in xy-plane: direct line integral ∮_{C} F·dr vs ∬_{S} curl F·n dS should match ⇒ often asked as verification problem (next topic).

(d) Need orientation: if C clockwise when seen from tip of n, sign flips.

(e) If F=∇φ, then curl zero ⇒ ∮=0 for any closed C that bounds a surface where curl continuous — re-proves conservative.

5. Relation to Green

Green's is Stokes in the plane: $S$ flat in $xy$, $n=k$, $(∇×F)·k = Q_x - P_y$.

6. Exam Points

• State Stokes' theorem — 3-5 marks, almost certain.
• Evaluate $∮_{C}F·dr$ via $∬_{S}(∇×F)·n dS$ for circle/hemisphere — 7 marks, very frequent.
• Verify for given $F, S, C$ — next topic's verification problems often use Stokes — 10 marks, very frequent (compulsory).
• Show $∮_{C}F·dr=0$ if $∇×F=0$ — 3 marks.