Question
Asked Dec 7, 2019
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Use Stokes' Theorem to evaluate 

 F · dr

 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).
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Expert Answer

Step 1

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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). Fdr = [[ curlF - ndS. According to Stokes' theorem, i k = -2zi – 2xj-2 yk . дz Obtain curlF as ôy x+ y y+z z+x² The vertices (3,0,0), (0,3,0), and (0,0,3) belong to the plane x+ y+z=3

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Step 2

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The projection of the plane x+y+z=3 onxy-plane is x+y=3. Thus, the surface S becomes {(x,y)|0<x<3,0<y<3-x}.

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Step 3

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-33-x [F- dr = (-2(3 – x – y)i– 2xj– 2yk)·(i+j+k)dxdy Jo Jo .3 3-x - "(-6) xdy -LI-6y" dx - ,(-18+ 6x )kr 3-x %3D That is, F dr =[-18x+3x² = -54+27 = -27

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Tagged in

Math

Calculus