THE UNIVERSITY OF CALGARY DEPARTMENT OF MATHEMATICS AND STATISTICS FINAL EXAMINATION MATH 353 (L60) Z 2. cos d. p 2 sin ddd =

Transcription

1 THE UNIVEITY OF CALGAY EPATMENT OF MATHEMATIC AN TATITIC FINAL EXAMINATION MATH 5 (L6) UMME, TIME: hours. Evaluate the integral + y ddy where is the region between two irles + y and + y 9; on the right from the y ais. f + y 9; g use olar oord.then f r 9; + y ddy r os rdrd r ; g os d rdr [sin ] h i r 9 5:. Evaluate zydv + y + z ;where f(; y; z) ; z ; y ; + y + z g : use sherial oord. ; ; [; ] ; : zydv + y + z os sin sin sin ddd d 8 : sin d os sin d [ os ] h i sin. erive the formula for the surfae area of a shere + y + z for any > : (A ) : the surfae ould be desribed as z y for (; y) f + y g then n (rz; )! y ; y y ;

2 q and knk + y so A A + (z > ) y y + d + y ddy r ; where f r ; g;. Use Green s Theorem to laulate y knk ddy r drd (olar) r where F ( y; y ) and the urve is boundary of the ellise + y ; oriented ounterlokwise. for any smooth F (F ; F ) by Green s Theorem i h(f ) (F ) y ddy; where is inside so for our eld (F ) (F ) y y and f + y g and [y ] ddy by modi ed olar oord. r os ; y r sin we know that + y r and ddy rdrd thus 8 r dr [y ] ddy (sin os )d 8 +os ( 5 )d r sin r os rdrd h i r : ( 5 os )d 5. how that for any smooth (i.e. with ontinuous seond order artials) onservative vetor eld F of variables div F ; where is a otential of F and is Lalae yy zz : we know that F r ( ; y ; z ) and div F (F ) + (F ) y + (F ) z

3 together div F ( ) + ( y ) y + ( z ) z + yy + zz : 6. Find the u of F (yz; y; z + ) outward from the surfae -art of the araboloid z y above the y lane (a) inluding the bottom; (b) eluding the bottom. l f z y ; (; y) g...lateral surfae and b f z ; (; y) g...bottom where f + y g for a) the surfae is losed so we an use Gauss Theorem u F d divf ddydz where is inside f z y ; (; y) g divf (F ) + (F ) y + (F ) z (yz) + (y) y + (z + ) z yz + + z F d [ + (y + ) z] ddydz + + z () y (sine the integrand funtion is odd in and set is symmetrial in ) (sine the integrand funtion is odd in y and set is symmetrial in y ) ( y ) ddy ( olar) ( r ) rdrd d for b) h ( r ) rdr ( 6 r ) i is not losed but we an use art a) sine l F d F d b F d 6? so we have to alulate the u through the bottom b f z ; (; y) g n (; ; ) and on b F (yz; y; z + ) z (; y; ) 6 : ddy

4 F d F n ddy b +os d therefore 7. Evaluate h i r F d 6 l + 76 : ddy os d r rdr where F (e y ; + e y ; e z ) and is losed urve, oriented ounterlokwise fz yg \ f + y g. sine the urve is losed we an use tokes Theorem with fz y; for + y g; and uward n ( rz; ) ( y; ; ) + + we z e y + e y e z (; ; + y ) thus urlf d (; ; + y ) ( y; ; ) ddy f +y g ( + y ) ddy r dr : f +y g 8. Evaluate where F (y; y; z) and fz yg \ f + y g between A ( ; ; ) and (; ; ) : the urve is not losed nor the eld is onservative so we have to nd a arametrization of from the ylinder os t; y sin t and from z y we have r (t) os t; sin t; os t sin t os t; sin t; sin t now for A t ; and for t r (t) sin t; os t; os t and the eld on

5 F (y; y; z) sin t os t; sin t; sin t then F r sin t os t + sin t + sin t F r dt h : sin t i + os t sin t os t + sin t + sin t dt [os t] + + : 5