z = x Let be the curve of intersection of the cylinder x ^ 2 + y ^ 2 = 1 and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate int S curl F* dS if F = yi + zj + 2xk.

CMIIh

CMIIh

Answered question

2021-03-08

z = x Let be the curve of intersection of the cylinder x2+y2=1 and the plane , oriented positively when viewed from above . Let S be the inside of this curve , oriented with upward -pointing normal . Use Stokes ' Theorem to evaluate ScurlFdSifF=yi+zj+2xk.

Answer & Explanation

lamanocornudaW

lamanocornudaW

Skilled2021-03-09Added 85 answers

Step 1
Stock's Theorem:-
CF.drcurl(F)ds
Given that a curve say D (as name not mention in problem) of intersection of the cylinder x2+y2=1 and the plane z=x and F=y i+z j+2x k
Also, S be inside of this curve, oriented with upward-pointing normal, parameterize of this curve is
r=<x,y,z<x,y,x> as z = z
r=cos0i+sin0j+cos0k
since, we know x=1.cos0=cos0,y=1.sin0=sin0,z=x=cos0 as radius =1
Here, 002π(asx2+y2=12)
then,
r=<cos0,sin0,cos0>
drd0=r=<sin0,cos0,sin0>
F(x,y,z)=yi+zj+2xk
F(r(0))=sin0i+cos0j+2cos0k
Step 2
Now,
DFdr=02πF(r(0))drd0d0
DFdr=02π<sin0,cos0,2cos0><sin0,cos0,sin0>d0
DFdr=02π(sin20+cos202sin0cos0)d0
Note: cos20sin20=cos20,sin20=2sin0cos0
DFdr=02π(cos20sin20)d0=[sin202+cos202]02π
Note:

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