Statement of Green's

Statement of Green's

1. Statement — Green's Theorem in the Plane

Let $C$ be a positively oriented (counter-clockwise), piecewise smooth, simple closed curve in the $xy$-plane, and $D$ the region it encloses. If $P(x,y), Q(x,y)$ have continuous first partials on an open region containing $D$, then:

The line integral is taken once around $C$ in the positive sense (region always to left as you walk). This converts a closed line integral (circulation) to a double integral of $curl F$'s $k$ component ($F = P i+Q j$ ⇒ $∇×F = (Q_x -P_y)k$).

2. Two Special Forms

• Tangential (circulation) form above.
• Normal (flux) form:

(with $F=P i+Q j$) — the 2D divergence theorem (often called Green).

3. Why Useful

• Computes difficult closed line integrals by doing easier double integral (often $Q_x-P_y$ is constant/simple).
• Proves path independence when $Q_x-P_y=0$ on simply connected $D$.
• Gives area formula: choose $P=-y/2, Q=x/2$ ⇒ $Q_x-P_y=1$ ⇒

4. Worked Examples

(a) Evaluate ∮_{C} (-y dx + x dy) where C is circle x²+y²=a² counter-clockwise:
    P=-y, Q=x ⇒ Q_x=1, P_y=-1 ⇒ Q_x-P_y=2 ⇒ ∮ =∬_{D}2 dA=2·πa²=2πa² ✓ matches earlier direct 2πa².

(b) ∮_{C} x² y dx + x y² dy over triangle? Compute Q_x = y², P_y = x² ⇒ integrand y² - x² double integral.

(c) Area of ellipse x²/a²+ y²/b²=1: param x=a cos t, y=b sin t, t 0→2π ⇒ 1/2∮ x dy - y dx =1/2∫_{0}^{2π}ab dt =πab ✓

(d) Need orientation: if C clockwise, ∮ = -∬ (sign flips).

(e) If P,Q have singularities inside D (e.g., F= -y/(x²+y²) i+ x/(x²+y²) j), Green fails unless punctured — ∮ may be 2π not zero, showing hole matters.

5. Conditions and Pitfalls

• $C$ must be closed, simple, positively oriented; $P,Q$ must be $C¹$ on $D$ (no singularities inside).
• If $D$ multiply connected (hole), $∮_{outer} -∮_{inner}=∬$ (subtract holes).
• Don't mix $Pdx+Qdy$ with $Pdy - Qdx$ forms.

6. Exam Points

• State Green's theorem — 3-5 marks, almost certain.
• Evaluate $∮_{C} Pdx+Qdy$ via $∬_{D}(Q_x-P_y)dA$ for circle/rectangle/triangle — 7 marks, very frequent.
• Area via $1/2∮ x dy -y dx$ — 5 marks, frequent.
• Show failure when singularity inside — 5 marks, occasional.