6.2 Green's Theorem in the Plane

Green's Theorem bridges line integrals and double integrals, simplifying complex calculations. It's a powerful tool for finding areas, work, and flux in vector fields, making it essential for understanding multivariable calculus.

The theorem's applications are vast, from calculating enclosed areas to analyzing fluid dynamics. Mastering its use and understanding curve orientation are key skills for tackling advanced problems in physics and engineering.

Green's Theorem Fundamentals

Application of Green's Theorem

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• Green's Theorem connects line integral around simple closed curve to double integral over enclosed region $\oint_C (P dx + Q dy) = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA$
• Components include $C$ (simple closed curve, counterclockwise), $D$ (enclosed region), $P$ and $Q$ (functions with continuous partial derivatives)
• Application steps:
  1. Identify closed curve $C$ and enclosed region $D$
  2. Determine $P$ and $Q$ from given line integral
  3. Calculate partial derivatives $\frac{\partial Q}{\partial x}$ and $\frac{\partial P}{\partial y}$
  4. Set up and evaluate double integral over $D$

Area calculation with Green's Theorem

• Special case uses $P = -y$ and $Q = x$ in formula $Area = \frac{1}{2} \oint_C (x dy - y dx)$
• Calculation steps:
  1. Parameterize closed curve $C$
  2. Express $x$ and $y$ in terms of parameter
  3. Set up line integral using area formula
  4. Evaluate integral to find enclosed area (ellipse, circle)

Orientation of closed curves

• Counterclockwise (positive) vs clockwise (negative) orientation affects line integrals
• Counterclockwise uses Green's Theorem as stated, clockwise requires negation or curve reversal
• Determine orientation:
  • Parametric method observes travel direction as parameter increases
  • Right-hand rule points thumb along positive z-axis, fingers curl in curve direction
• Multiple curves require counterclockwise outer curve, clockwise inner curves (holes)

Work and flux using Green's Theorem

• Work calculations:
  • Force field $\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}$
  • Work formula $W = \oint_C \mathbf{F} \cdot d\mathbf{r} = \oint_C (P dx + Q dy)$
  • Apply Green's Theorem to convert line integral to double integral
• Flux calculations:
  • Vector field $\mathbf{F}(x,y) = P(x,y)\mathbf{i} + Q(x,y)\mathbf{j}$
  • Flux formula $Flux = \oint_C \mathbf{F} \cdot \mathbf{n} ds = \oint_C (P dy - Q dx)$
  • Use Green's Theorem with modified integrand
• Problem-solving steps:
  1. Identify work or flux problem
  2. Determine force or vector field components
  3. Set up appropriate line integral
  4. Apply Green's Theorem for double integral conversion
  5. Evaluate resulting double integral (gravitational field, fluid flow)