Chapter 9: Vector Calculus
Section 9.10: Green's Theorem
Use Green's theorem to calculate the area inside a circle of radius a.
Without loss of generality, the circle can be centered at the origin, so it will have the Cartesian representation x2+y2=a2. To continue working in Cartesian coordinates, obtain y±=±a2−x2. Apply the "formula" A=−∳Cy dx to obtain
=−∫a−ay+ dx+∫−aay− dx
=−−∫−aay+ dx − ∫−aay+ dx
Now an antiderivative for y+ is x y++a2 arctanx/y+/2. Since this will then be evaluated at the endpoints where y+vanishes, it is better to use the two-argument form of the arctangent function. Consequently, the computed value for A will be
An alternative evaluation for the integral of y+starts with the trigonometric substitution x=a sinu.
Maple Solution - Interactive
Write the equation of the circle.
Context Panel: Solve≻Obtain Solutions for≻y
Context Panel: Assign to a Name≻Y
x2+y2=a2→solutions for ya2−x2,−a2−x2→assign to a nameY
Calculus Palette: Definite-integral template
Context Panel: Evaluate and Display Inline
Context Panel: Simplify≻Assuming Positive
−∫a−aY1 ⅆx+∫−aaY2 ⅆx = csgn⁡a⁢a2⁢π→assuming positivea2⁢π
Alternatively, 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
Calculus - Vector≻Integration≻Flux≻2-D≻Through a Circle
Flux through a Circle
Select Coordinate SystemCartesian [x,y]Cartesian - otherpolarbipolarcardioidcassinianelliptichyperbolicinvcassinianlogarithmiclogcoshparabolicrosetangent
Applying the factor of 1/2 to the calculated value of the flux results in the expected π a2.
Maple Solution - Coded
Use the solve command to obtain y±.
Apply the top-level int command under the assumption that a is positive.
−intY1,x=a..−a+intY2,x=−a..a assuming a>0 = a2⁢π
The alternate solution based on a direct calculation of flux is contained in Table 9.10.3(a).
Install the Student VectorCalculus package.
Define F with the VectorField command.
Calculate half the flux with the Flux command
FluxF,Circle0,0,a/2 = π⁢a2
Table 9.10.3(a) Calculation of area with the divergence-form of Green's theorem
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