Use Stokes' Theorem to evaluate int int_S CURL f * dS. F(x,y,z)=x^2y^3zi+sin(xyz)j+xyzk, S is the part of the cone y^2=x^2+z^2 that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.

iohanetc 2021-03-09 Answered
Use Stokes' Theorem to evaluate SCURLfdS.
F(x,y,z)=x2y3zi+sin(xyz)j+xyzk,
S is the part of the cone y2=x2+z2 that lies between the planes y = 0 and y = 2, oriented in the direction of the positive y-axis.
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liingliing8
Answered 2021-03-10 Author has 95 answers
Solution:
The vector field is F(x,y,z)=x2y3zi+sin(xyz)j+xyzk
Find: ScurlF·dS
Then, the double integral becomes ScurlF·dS=CF·dr
The surface of a cone is y2=x2+z2.
At y=2,x2+z2=4.
Let the parametric equation is r(t)=<2cost,2,2sint>0t2π and its derivative is r(t)=<2sint,0,2cost>.
Then, the vector field becomes
F(r(t))==(2cost)2(2)3(2sint)i+sin(8sintcost)j+8sintcostk(64cos2tsint)i+sin(8sintcost)j+8sintcostk
Step 2
Obtain the integal:
CF·dr=02π0<64cos2tsint,sin(8sintcost),8sintcost><2sint,0,2cost>02π(128cos2tsin2t+16sintcos2t)dt32π
Therefore, ScurlF·dS=32π.
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