Green's Theorem Calculator
Relates a line integral around a closed curve to a double integral over the region it encloses.
Formula first
Overview
Green's Theorem establishes a fundamental connection between the line integral around a simple closed curve and the double integral over the plane region it encloses. It is essentially a two-dimensional version of the Stokes' Theorem and is used to relate local rotation or circulation in a vector field to the net curl over an area.
Symbols
Variables
\text{Concept-only} = Note
Apply it well
When To Use
When to use: Apply this theorem when evaluating a line integral over a closed, piecewise-smooth curve in the xy-plane where the area integral of the curl is easier to compute. It requires the component functions L and M to have continuous first-order partial derivatives throughout the region bounded by the curve.
Why it matters: It is essential for calculating work and circulation in physics and fluid dynamics without needing to parameterize complex boundary paths individually. It also provides a mathematical basis for using line integrals to calculate the area of irregular shapes, which is the operational principle behind the planimeter.
Avoid these traps
Common Mistakes
- Using for open curves.
- Wrong sign (clockwise orientation).
One free problem
Practice Problem
Evaluate the line integral ∮_C (y² dx + x² dy) where C is the boundary of the rectangle defined by 0 ≤ x ≤ 2 and 0 ≤ y ≤ 3, oriented counter-clockwise.
Solve for:
Hint: Convert the line integral into a double integral of the expression (∂M/∂x − ∂L/∂y) over the rectangular region.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Calculus: Early Transcendentals by James Stewart
- Vector Calculus by Jerrold E. Marsden and Anthony J. Tromba
- Wikipedia: Green's theorem
- Stewart, Calculus: Early Transcendentals
- Halliday, Resnick, and Walker, Fundamentals of Physics
- Bird, Stewart, and Lightfoot, Transport Phenomena
- Britannica, Green's theorem
- Wikipedia, Green's theorem