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Let F = < a?eUZ, xe*z, z?e"y > . Use Stokes' Theorem to evaluate curlF · dS , where S is the hemisphere x? + y? + z2 = 9, z > 0, oriented upwards

Let F = < a?eUZ, xe*z, z?e"y > . Use Stokes' Theorem to evaluate curlF · dS , where S is the hemisphere x? + y? + z2 = 9, z > 0, oriented upwards

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let F = < x?eUZ, xe²², z²e#y > . curlF · d§ , where Use Stokes' Theorem to evaluate S is the hemisphere x? + y? + z? = 9, z > 0, oriented upwards
Transcribed Image Text:Let F = < x?eUZ, xe²², z²e#y > . curlF · d§ , where Use Stokes' Theorem to evaluate S is the hemisphere x? + y? + z? = 9, z > 0, oriented upwards
Expert Solution
Step 1

Given that, F=x2eyz,xexz,z2exy

Given hemisphere that S:x2+y2+z2=4,z≥0

From stoke's theorem,

∬Scurl  F.dS=∫CF.dr

A parametrization is:

x=rcosθ⇒x=2cosθy=rsinθ⇒x=2sinθz=0

So, F=4cos2θ,2cosθ,0 and r(θ)=2cosθ,2sinθ,0 then, dr=-2sinθ,2cosθ,0

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