Solutions to Test #2 (Kawai) MATH 2421

Transcription

1 Solutions to Test # (Kawai) MATH 4 (#) Each vector eld deicted below is a characterization of F (; ) hm; Ni : The directions of all eld vectors are correct, but the magnitudes are scaled for ease of grahing. Match each clue with its corresonding grah. It is ossible that a grah could be used more than once... (i) Which eld most resembles F (; ) h; answer is (b). i? This is the clockwise eld. The correct (ii) Which eld aears to have negative divergence at the origin? The eld vectors oint toward the origin. The correct answer is (d). (#) Which ONE statement is alwas TRUE? The correct answer is (c). (a) rf : This is an unde ned oeration. We cannot take the gradient of a vector eld. (b) r F : Not all elds have zero curl. (c) If f is a -variable di erentiable function, then r (rf) h; ; i : YES. Gradient elds are ALWAYS conservative. (d) If F is conservative, then r F : Mabe. The onl guarantee is that the curl is the zero vector. * (#) If F (; ) + ; + + ; ; what is the divergence, r F? Quotient Rule! [Multil to and bottom b + :] + + () B smmetr, we must "! ( + ) + # ( + ) ( + ) : and their sum is r F ( + ) + ( + ) + ( + ) ( + ) + : The correct answer is (b).

2 (#4) If our surface S is the uer hemishere z f (; ) ; then which eression below is equivalent to the surface area di erential, ds? v q! u ds + (f ) + (f ) da t + +! da s + s + da + + r da da r : The correct answer is (b). (#) Reverse the order of integration and evaluate. We currentl have a horizontal rst region: d d left! right :! :! : This is the right triangle BELOW : Thus, if we go vertical rst, we have d d lower! uer :! :! : Inner: Outer: u + ; du d: + d d??? + d d ln () : + d + : + d ) du u ln juj + C ln + ln () ln () ln () :X

3 (#6) Find the mass of the uer solid unit shere + + z ; in OCTANT I ONLY; if the densit function is (; ; z) z + + z : Hint: Sherical. u-substitution. The uer hemishere limits are: sin () d d d :! :! :! : The densit function is Thus, we have z + + z cos () cos () : m cos () sin () d d d 4 sin () cos () d d d mass units. All the limits of integration are constants and the integrand is factorable.!! sin 4 () d sin () cos () d d I (#7) Find the F dra around C [once around, counterclockwise] ; C the boundar of the region + 4: The uid velocit vector eld is F (; ) ; e cos() : Hint: cos (u) du u Green s Theorem. We have the circulation densit: + sin (???) 4 + C N M r cos () : [Hence, the hint.] We must evaluate this over the circular region: r:! :! : :X

4 [All the limits of integration are constants. r cos () The integrand is factorable.] r dr cos () d sin () 4 + 4: 4 In general, the circulation around C is clockwise since the circulation number is negative. (#8) We alread know that F (; ) h; + cos () + sin ()i is conservative everwhere. Eas-ish integration b arts??? (a) Use the aroriate test to show that F is conservative. (b) Find a otential function (; ) for F: Md d + () Nd N : M ) N M :X ( + cos () + sin ()) d + ( sin () + cos ()) cos () + () + sin () + () : We collect one co of each term. (; ) + sin () + C: (c) Find the work accomlished b F when we move a article (along an ath) from (; ) to (; ) : B the Fundamental Theorem of Line Integrals (FTLI), we have F dr (; ) (; ) C + sin ( + sin ()) 4 + : (#9) Find the work accomlished b the non-conservative force eld F (; ) ; from (; ) to (6; ) along the curve sin () : Eas arameterization. r (t) ht; sin (t)i ) v (t) h; cos (t)i : Eas u-sub. The arameterization is 4

5 6 sin (t) ; sin (t) h; cos (t)i dt 6 u sin (t) ; du cos (t) dt: [for that second art...] [cos (t)] 6 sin 6! (t) + + sin (t) + sin (t) cos (t) dt : (#) Let f (; ) : We alread know that (; ) is a critical oint. (a) Find the other TWO critical oints. Show all work! When does f f? f 4 4 ) f 4 4 ) : Substitute. 9 ) 9 8 ) ; : Since ; we alread know that (; ) is a critical oint, and the other two are (; ) and ( ; ) : (b) Now run the Second Partials Test on the critical oint (; ) and classif it as min/ma/saddle. f ; f ; f 4: At the oint (; ) ; we have d f f (f ) ( ) ( ) 4 +8: Since f and f are both negative, this must be a relative MAXIMUM oint. (#) Let F (; ; z) he ; z; cos ()i : Evaluate the curl, r F: r F hp N z ; M z P ; N M i h sin () ; ( sin ()) ; z e i h sin () ; sin () ; z e i : (#) Suose we have one-half of a circle in Quadrant I. You should have memorized the olar form of this circle: r sin () ; : r:! sin () :! : Write down, but DO NOT EVALUATE this double integral using olar coordinates: + sin() da r [Ugl reduction of ower roblem.] R sin() r 4 dr d 6 7 :

6 (#) Suose S is the ortion of the ellitic araboloid z f (; ) + above the unit circle region + : Suose the densit function is ( + z) : This should look susiciousl familiar. Find the mass of S: The densit function is + z r : The surface area di erential is ds q + (f ) + (f ) da q + () + () da + + da + r : The region R in the -lane is: r:! :! : m 4 : + r + r + r Inner: u + r ; du r dr ) r dr () du: + r r dr ) h u du + r i u + C u + C The Outer integral multilies the revious result b : 4 : 6