Q4] Verify Divergence theorem of the regionx² + y² + (z-1)² = 9 and 1 ≤ z ≤ 4 where051 YonifuV which is bounded byA=xi+yj+(z − 1)k.

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Answered: Q4] Verify Divergence theorem of the…

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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Q4] Verify Divergence theorem of the region
x² + y² + (z-1)² = 9 and 1 ≤ z ≤ 4 where
051 Yonifu
V which is bounded by
A=xi+yj+(z − 1)k.

Expert Solution

Step 1: To find the flux of F across S:

$L e t space S equals S subscript 1 union S subscript 2 comma space w h e r e space S subscript 1 space i s space i s space t h e space h e m i s p h e r e space o f space r a d i u s space 3 space a n d space c e n t e r space a t space left parenthesis 0 comma 0 comma 1 right parenthesis a n d space S subscript 2 space i s space t h e space d i s k space x squared plus y squared less or equal than 9 space i n space z equals 1.$

$C a l c u l a t i o n space f r o m space S subscript 1 colon$

$P a r a m e t r i z e space S subscript 1 colon space r with rightwards arrow on top left parenthesis ϕ comma theta right parenthesis equals left parenthesis 3 sin ϕ cos theta comma 3 sin ϕ sin theta comma 3 cos ϕ plus 1 right parenthesis space a n d F with rightwards arrow on top left parenthesis r with rightwards arrow on top left parenthesis ϕ comma theta right parenthesis right parenthesis equals left parenthesis 3 sin ϕ cos theta comma 3 sin ϕ sin theta comma 3 cos ϕ right parenthesis comma space w h e r e space ϕ element of left square bracket 0 comma straight pi over 2 right square bracket space a n d space theta element of left square bracket 0 comma 2 straight pi right square bracket H e n c e comma open parentheses r with rightwards arrow on top subscript ϕ cross times r with rightwards arrow on top subscript theta close parentheses equals left parenthesis 9 sin squared ϕ cos theta comma 9 sin squared ϕ sin theta comma 9 sin ϕ cos ϕ right parenthesis T h e r e f o r e comma space F with rightwards arrow on top left parenthesis r with rightwards arrow on top left parenthesis ϕ comma theta right parenthesis right parenthesis. open parentheses r with rightwards arrow on top subscript ϕ cross times r with rightwards arrow on top subscript theta close parentheses equals left parenthesis 3 sin ϕ cos theta comma 3 sin ϕ sin theta comma 3 cos ϕ right parenthesis. left parenthesis 9 sin squared ϕ cos theta comma 9 sin squared ϕ sin theta comma 9 sin ϕ cos ϕ right parenthesis equals 27 sin ϕ. H e n c e space F l u x space o f space F with rightwards arrow on top space a c r o s s space S subscript 1 space i s integral subscript S subscript 1 end subscript integral space F with rightwards arrow on top. d S with rightwards arrow on top subscript 1 equals integral subscript D subscript 1 end subscript integral F with rightwards arrow on top left parenthesis r with rightwards arrow on top left parenthesis ϕ comma theta right parenthesis right parenthesis. open parentheses r with rightwards arrow on top subscript ϕ cross times r with rightwards arrow on top subscript theta close parentheses equals 27 integral subscript 0 superscript straight pi over 2 end superscript integral subscript 0 superscript 2 straight pi end superscript sin ϕ d theta d ϕ equals 54 straight pi$

$C a l c u l a t i o n space f o r space S subscript 2 colon P a r a m e t r i z e space S subscript 2 colon space t with rightwards arrow on top left parenthesis r comma theta right parenthesis equals left parenthesis r cos theta comma r sin theta comma 1 right parenthesis space a n d F with rightwards arrow on top left parenthesis t with rightwards arrow on top left parenthesis r comma theta right parenthesis right parenthesis equals left parenthesis r cos theta comma r sin theta comma 0 right parenthesis comma space w h e r e space r element of left square bracket 0 comma 3 right square bracket space a n d space theta element of left square bracket 0 comma 2 straight pi right square bracket H e n c e comma open parentheses t with rightwards arrow on top subscript r cross times t with rightwards arrow on top subscript theta close parentheses equals left parenthesis 0 comma 0 comma r right parenthesis T h e r e f o r e comma space F with rightwards arrow on top left parenthesis t with rightwards arrow on top left parenthesis r comma theta right parenthesis right parenthesis. open parentheses t with rightwards arrow on top subscript r cross times t with rightwards arrow on top subscript theta close parentheses equals left parenthesis r cos theta comma r sin theta comma 0 right parenthesis. left parenthesis 0 comma 0 comma r right parenthesis equals 0. H e n c e space F l u x space o f space F with rightwards arrow on top space a c r o s s space S subscript 2 space i s space 0 H e n c e space t h e space t o t a l space f l u x space i s space left parenthesis 54 straight pi plus 0 right parenthesis equals 54 straight pi$

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Q4] Verify Divergence theorem of the region x² + y² + (z − 1)² = 9 and 1 ≤ z ≤ 4 where 051 Vouifu V which is bounded by A=xi+yj+(z −1)k.