Respuesta :
The integral || curl F || dĞ is equal to -2. The correct answer is therefore option D that is -87.
To evaluate the integral || curl F || dĞ using Stoke's Theorem, we need to find a vector field F and a surface S such that the boundary of S is a closed curve C and the integral is equal to the flux of the curl of F through S. Using the given information, we can find the components of F as:
F = (-y, x, r)
The curl of F is given by:
curl F = (0, 0, 1)
The surface S is the part of the sphere r2 + y2 + z2 = 4 that lies above the cy-planes, oriented upward. This surface can be parameterized by:
x = rcosφcosθ
y = rcosφsinθ
z = rsinφ
where 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π.
The boundary of S is the curve C, which is the intersection of the sphere and the cy-plane. This curve can be parameterized by:
x = rcosθ
y = rsinθ
z = 0
where 0 ≤ θ ≤ 2π.
Using Stoke's Theorem, the integral || curl F || dĞ is equal to the flux of curl F through S, which is given by:
∫|| curl F || dĞ = ∫curl F · ndC
where n is the unit normal vector to the surface S.
Substituting the expressions for curl F and the parameterizations of S and C, we get:
∫|| curl F || dĞ = ∫(0, 0, 1) · (cosφcosθ, cosφsinθ, sinφ)dC
= ∫(sinφ)dC
= ∫sinθdθ
= [-cosθ]θ=0θ=2π
= -2
Learn more about Stoke's theorem at:
brainly.com/question/13105453
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