20.1 Green's theorem in the plane

Green's theorem connects line integrals and double integrals, showing how they're related for closed curves in a plane. It's a powerful tool that simplifies calculations and provides insights into vector fields, making it easier to solve problems in physics and engineering.

This theorem is part of a broader family of integral theorems in vector calculus. It sets the stage for more advanced concepts like Stokes' theorem and the divergence theorem, which extend these ideas to three dimensions and beyond.

Line Integrals and Double Integrals

Integral Definitions

Top images from around the web for Integral Definitions

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

1 of 3

Top images from around the web for Integral Definitions

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  Double Integrals over General Regions · Calculus View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

• 

  HartleyMath - Double Integrals over Rectangular Regions View original

  Is this image relevant?

1 of 3

• Line integral evaluates a function along a curve or path in a plane or space
  • Computes the work done by a force along a path
  • Can also calculate the mass of a wire or flow rate of a fluid through a pipe
• Double integral integrates a function of two variables over a region in a plane
  • Generalizes the concept of area under a curve to a function of two variables
  • Useful for calculating volume under a surface, mass of a flat plate, or electric charge on a surface

Closed Curves and Vector Fields

• Closed curve is a path that starts and ends at the same point without crossing itself
  • Examples include circles, ellipses, and any simple polygon like a triangle or square
• Vector field assigns a vector to each point in a subset of space
  • Common in physics to represent forces, velocities, or electromagnetic fields
  • Examples: gravitational field, electric field, fluid flow velocity field

Green's Theorem

Statement and Orientation

• Green's Theorem relates a line integral around a simple closed curve to a double integral over the region bounded by the curve
  • Converts a line integral to an equivalent double integral, which is often easier to compute
  • Useful for evaluating line integrals around complicated curves
• Theorem requires the curve to have a "positively oriented boundary"
  • Means traversing the boundary keeps the enclosed region always to the left
  • Equivalent to moving along the boundary in a counterclockwise direction
  • Important because reversing orientation changes the sign of the line integral

Applications

• Calculating work done by a force field around a closed path
  • Example: work required to move an object around a closed loop in a gravitational field
• Finding the area enclosed by a closed curve
  • Green's theorem simplifies this to a double integral, avoiding parameterization of the curve
• Evaluating flux of a vector field across a closed curve
  • Flux measures how much of a vector field is crossing a boundary
  • Useful in electromagnetism for calculating electric flux through a closed loop

Vector Calculus Concepts

Curl and Divergence

• Curl is a vector operator that measures the infinitesimal rotation of a vector field
  • Tells how much the vector field tends to circulate around a point
  • A vector field with zero curl is called conservative or irrotational
  • Example: velocity field of a vortex or whirlpool has non-zero curl
• Divergence is a scalar operator that measures the infinitesimal flux of a vector field
  • Determines the tendency of a vector field to originate from or converge to a point
  • Positive divergence indicates net outward flux, negative is net inward flux
  • Example: electric field of a positive point charge has positive divergence

Significance in Vector Fields

• Curl and divergence characterize important properties of vector fields
  • Conservative fields (zero curl) have path-independent line integrals useful in physics
  • Incompressible flows, like some fluid flows, have zero divergence
• Green's theorem can be expressed in terms of curl and divergence
  • Relates line integral of a vector field to double integral of its curl and divergence
  • Provides deeper insight into the meaning and applications of Green's theorem
• Many other important theorems in vector calculus generalize Green's theorem
  • Example: Stokes' theorem relates surface integral of curl to line integral around boundary
  • Divergence theorem relates volume integral of divergence to surface integral of flux