Use Stokes' Theorem to evaluate int_C F * dr where C is oriented counterclockwise as viewed from above. F(x,y,z)=(x+y^2)i+(y+z^2)j+(z+x^2)k, C is the triangle with vertices (3,0,0),(0,3,0), and (0,0,3).

sagnuhh

sagnuhh

Answered question

2021-02-25

Use Stokes' Theorem to evaluate CFdr where C is oriented counterclockwise as viewed from above.
F(x,y,z)=(x+y2)i+(y+z2)j+(z+x2)k,
C is the triangle with vertices (3,0,0),(0,3,0), and (0,0,3).

Answer & Explanation

brawnyN

brawnyN

Skilled2021-02-26Added 91 answers

Solution:
The vector field is F(x,y,z)=(x+y2)i+(y+z2)j+(z+x2)k.
The equation of the plane is x+y+z=3.
Consider z=g(x,y)=3xy.
Use Stokes’ Theorem and get the surface integral set up.
CF×dr=Scurl F×dS=ScurlF× fdA
Obtain the curl of F:
curlF=[ijkxyzx+y2y+z2z+x2]=<2z,2x,2y>
Conclusion:
Then, f(x,y,z)=z3+x+y.f=<1,1,1>
Plug z=0,x+y=3y=3x
CFdr=S2z,2x,2y<1,1,1>dA
=0303x2(z+x+y)dydx
=60303xdydxz+x+y=3
=603[3x]dx
That is, CFdr=6[3xx22]=27.

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