Respuesta :
Green's theorem in vector calculus connects a line integral around a simple closed curve C to a double integral over the plane area D limited by C.
What is Green's Theorem?
Green's theorem in vector calculus connects a line integral around a simple closed curve C to a double integral over the plane area D limited by C. It is a two-dimensional variant of Stokes' theorem.
The path of the particle is positively oriented so we can use Green's theorem to find the work done in moving particle along C.
The work done is given by [tex]\oint_{c}\vec F \cdot \vec{dr}[/tex]
Here, [tex]\vec F[/tex](x,y)=<x+y, y²-x>
Then [tex]\oint_{c}\vec F \cdot \vec{dr} = \oint_{c}(x+y)dx+(y^2-x)dy[/tex]
In comparison with [tex]\oint_{c}Pdx+Qdy[/tex] what we have,
P= x+y and Q=y²-x
Then [tex]\dfrac{\partial P}{\partial y} = 1[/tex] and [tex]\dfrac{\partial Q}{\partial x} =-1[/tex]
Using the Green's Theorem,
[tex]\oint_{c}Pdx+Qdy = \iint_{D} \left ( \dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right )dA[/tex]
Here D in polar Co-ordinates is given by,
D={(r,θ) : 0 ≤ r ≤ 7, 0 ≤ θ ≤ π}
Then
[tex]\oint_{c}(x+y)dx+(y^2-x)dy\\\\=-\iint_D 2dA\\\\=-\int_{\pi}^{0}\int_{0}^7 2r drd\theta\\\\=-49\pi[/tex]
Learn more about Green's Theorem:
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