Integral theorems
Objective. Apply Green, Stokes and divergence theorems.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Integral theorems connect line, surface and volume integrals through derivative operators. Green's theorem relates a line integral around a closed plane curve C to a double integral: ∮_C P dx + Q dy = ∬_R (∂Q/∂x − ∂P/∂y) dA. Stokes' theorem generalises this to surfaces in space using curl, and the Divergence theorem relates a closed surface integral to a volume integral of divergence: ∬_S F·n dS = ∭_V div F dV. For F = (x, y) around the unit circle, Green's theorem gives ∬_R (0 − 0) dA = 0, matching direct computation. These theorems convert hard boundary integrals into often simpler region integrals.
Worked example
Use Green's theorem to evaluate ∮_C (x² dx + xy dy) around the unit square with vertices (0,0), (1,0), (1,1), (0,1).
- 1Identify P = x², Q = xy, so ∂Q/∂x = y and ∂P/∂y = 0.
- 2Green's theorem gives ∮_C P dx + Q dy = ∬_R y dA over the unit square.
- 3Compute ∬_R y dA = ∫₀¹∫₀¹ y dx dy = ∫₀¹ y dy = 1/2.
- 4So the closed line integral equals the double integral value found.
Answer: 1/2
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Start practisingCommon mistake. Students apply Green's theorem with the curve traversed clockwise without negating the result; Green's theorem assumes a counterclockwise (positively oriented) boundary, so clockwise orientation requires a sign flip.
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