Question
Asked Dec 3, 2019
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Use Stokes' Theorem to evaluate 

curl F · dS.
F(x, y, z) = 6y cos(z) i + ex sin(z) j + xey k,

S is the hemisphere x2 + y2 + z2 = 25, z ≥ 0, oriented upward.
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Expert Answer

Step 1

From the given s...

F(x,y, z) 6y cos (z)i+esin (z)j+ xe'k.
The hemisphere S is r2
225,z20
Stoke's Theorem:curlFdS =Fdr
Consider the boundary curve C is the circlex2 + y2 z2 = 25,z 0, where the
hemisphere intersects the xy- plane. This boundary curve C must be obtained in the
counter-clockwise direction. Therefore, the vector equation of C is,
r(t) 5cos(-)i5sin(-t)
=5costi 5sinfj
Differentiate the equation with respect to t
(t)-5sinfi-5costj
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F(x,y, z) 6y cos (z)i+esin (z)j+ xe'k. The hemisphere S is r2 225,z20 Stoke's Theorem:curlFdS =Fdr Consider the boundary curve C is the circlex2 + y2 z2 = 25,z 0, where the hemisphere intersects the xy- plane. This boundary curve C must be obtained in the counter-clockwise direction. Therefore, the vector equation of C is, r(t) 5cos(-)i5sin(-t) =5costi 5sinfj Differentiate the equation with respect to t (t)-5sinfi-5costj

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