Use Stokes’ Theorem to evaluate int int curl F * dS.\ F(x,y,z)=2y cos zi+e^x sin zj+xe^y k, S is the hemisphere x^2+y^2+z^2=9, z>=0,ZSK oriented upward

ximblajy

ximblajy

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2022-08-20

Use Stokes’ Theorem to evaluate curlFdS. F(x,y,z)=2ycoszi+exsinzj+xeyk, S is the hemisphere
x2+y2+z2=9,z0,
oriented upward

Answer & Explanation

Branson Grimes

Branson Grimes

Beginner2022-08-21Added 9 answers

When applying the Stokes's theorem the first thing you need to do is to identify the correct boundary of the surface. In our case, this boundary is the circle of radius 3 at the xy plane. We immediately parametrize it as:
γ(t)=(3cost,3sint,0)γ(t)=(3sint,3cost,0)
Next, recall that the Stokes's theorem states that:
ScurlFdS=SFds=02πF(γ(t))γ(t)
The integral becomes:
02π18sin2tdt=18π
Result:
18π

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