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Greens Theorem Calculator

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Example
Created on 2024-06-20Asked by Jack Martinez (Solvelet student)
Use Green's Theorem to evaluate the line integral ∮C(y2 dx+x2 dy) \oint_C (y^2 \, dx + x^2 \, dy) where C C is the circle x2+y2=1 x^2 + y^2 = 1 .

Solution

To use Green's Theorem to evaluate the line integral ∮C(y2 dx+x2 dy) \oint_C (y^2 \, dx + x^2 \, dy) where C C is the circle x2+y2=1 x^2 + y^2 = 1 : Green's Theorem states: ∮C(P dx+Q dy)=∬D(∂Q∂x−∂P∂y) dA \oint_C (P \, dx + Q \, dy) = \iint_D \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \, dA For P=y2 P = y^2 and Q=x2 Q = x^2 : ∂Q∂x=2x \frac{\partial Q}{\partial x} = 2x ∂P∂y=2y \frac{\partial P}{\partial y} = 2y ∮C(y2 dx+x2 dy)=∬D(2x−2y) dA \oint_C (y^2 \, dx + x^2 \, dy) = \iint_D (2x - 2y) \, dA In polar coordinates, x=rcos⁡θ x = r \cos \theta and y=rsin⁡θ y = r \sin \theta , dA=r dr dθ dA = r \, dr \, d\theta , and the region D D is the disk 0≤r≤1 0 \leq r \leq 1 , 0≤θ≤2π 0 \leq \theta \leq 2\pi : ∬D(2x−2y) dA=∬D(2rcos⁡θ−2rsin⁡θ)r dr dθ \iint_D (2x - 2y) \, dA = \iint_D (2r \cos \theta - 2r \sin \theta) r \, dr \, d\theta =∫02π∫012r2(cos⁡θ−sin⁡θ) dr dθ = \int_0^{2\pi} \int_0^1 2r^2 (\cos \theta - \sin \theta) \, dr \, d\theta =∫02π(cos⁡θ−sin⁡θ)[2r33]01 dθ = \int_0^{2\pi} (\cos \theta - \sin \theta) \left[ \frac{2r^3}{3} \right]_0^1 \, d\theta =23∫02π(cos⁡θ−sin⁡θ) dθ = \frac{2}{3} \int_0^{2\pi} (\cos \theta - \sin \theta) \, d\theta Since ∫02πcos⁡θ dθ=0 \int_0^{2\pi} \cos \theta \, d\theta = 0 and ∫02πsin⁡θ dθ=0 \int_0^{2\pi} \sin \theta \, d\theta = 0 : 23∫02π(cos⁡θ−sin⁡θ) dθ=0 \frac{2}{3} \int_0^{2\pi} (\cos \theta - \sin \theta) \, d\theta = 0 Therefore: ∮C(y2 dx+x2 dy)=0 \oint_C (y^2 \, dx + x^2 \, dy) = 0 Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mason Clark on Solvelet
1. Use Green's theorem to evaluate the line integral ∮(P dx+Q dy)\oint (P \, dx + Q \, dy) along the boundary of the region RR, where P=x2+y2P = x^2 + y^2 and Q=−xyQ = -xy,2. Compute the circulation of the vector field F=yi−xj\mathbf{F} = yi - xj around the boundary of the region bounded by the circle x2+y2=4x^2 + y^2 = 4 using Green's theorem.,
DefinitionGreen's Theorem - Green's Theorem is a fundamental theorem in vector calculus that relates the line integral of a vector field over a simple closed curve to the flux of the vector field through the enclosed region. It relates the circulation around a curve to the flux through a region, allowing a complicated line integral to be turned into a more easily-computable area integral. Green's theorem has applications in the study of fluid flow, electromagnetism and conservative field in physics, engineering, and mathematics. Greens Theorem: If F=(P,Q) is a vector field and C is a postively oriented simple closed curve that encloses the region D then ∮C(Pdx+Qdy)=∬D(∂x∂Q​−∂y∂P​)dA
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