Chapter 9: Vector Calculus
Section 9.10: Green's Theorem
Use Green's theorem to calculate the area inside the ellipse x/a2+y/b2=1.
Write the ellipse in polar coordinates to obtain r2cos2t/a2+sin2t/b2=1, so
Use the "formula" A=∳Cx dy, with x=rt cost and
dy=ddtrt sint dt=r′t sint+rt cost dt
where, after simplification, r′t=a b b2−a2 costsinta2sin2t+b2cos2t3/2.
Putting this all together gives
A=∫02 πrtcostr′t sint+rt cost dt = π a b
Of course, the actual integration is complex enough to warrant the use of Maple, and even then, Maple must know that both a and b are positive before it can complete the calculation.
Maple Solution - Interactive
Recall that the "formula" A=12∳Cx dy−y dx derives from the divergence-form of Green's theorem, so that the line integral is the flux of the field F=x i+y j through the curve C. This flux is obtained with the following task template.
Should the "Clear All and Reset" button in the Task Template be pressed, all the data that has been input to the template will be lost. In that event, the reader should simply re-launch the example to recover the appropriate inputs to the template.
Tools≻Load Package: Student Vector Calculus
Context Panel: Assign to a Name≻R
a bb2cos2θ+a2sin2θ→assign to a nameR
Calculus - Vector≻Integration≻Flux≻2-D≻Through a Plane Curve
Flux through a Plane Curve
Select Coordinate SystemCartesian [x,y]Cartesian - otherpolarbipolarcardioidcassinianelliptichyperbolicinvcassinianlogarithmiclogcoshparabolicrosetangent
Select Coordinate SystemCartesianpolarbipolarcardioidcassinianelliptichyperbolicinvcassinianlogarithmiclogcoshparabolicrosetangent
The Simplify button has been pressed to simplify the flux integral. Since both a and b are positive, csgna/b=1, and applying the factor of 1/2 to the calculated value of the flux results in the expected π a b.
Maple Solution - Coded
Recall that the "formula" A=12∳Cx dy−y dx derives from the divergence-form of Green's theorem, so that the line integral is the flux of the field F=x i+y j through the curve C. This flux is obtained by the direct calculation contained in Table 9.10.5(a).
Install the Student VectorCalculus package.
Define F with the VectorField command.
Define the ellipse parametrically.
Calculate half the flux with the Flux command, aided by simplify and appropriate assumptions
FluxF,PathR,θ,θ=0..2 π,coords=polarr,θ/2 assuming a>0,b>0 = π⁢a⁢b
Table 9.10.5(a) Calculation of area with the divergence-form of Green's theorem
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