Math iscussio Problems Path Idepedece. Let be the striaght-lie path i R from the origi to (3, ). efie f(x, y) = xye xy. (a) Evaluate f dr. (b) Evaluate ((, 0) + f) dr. (c) Evaluate ((y, 0) + f) dr.. Let be the positively orieted uit circle i R cetered at the origi. (a) Evaluate y dx x dy. (b) Use the previous part to explai why (y, x) is t the gradiet of a fuctio. 3. For each of the followig vector fields, determie whether it is the gradiet of a fuctio. (a) (4x 4y + x, 7xy + l y) o R (b) (3x l x + x, x 3 /y) o R (c) (x 3 y, 0, z ) o R 3 4. For each of the followig, fid the fuctio f. (a) f(0, 0, 0) = 0 ad f = (x, y, z) (b) f(,, 3) = 4 ad f = (, 6, 7) (c) f(,, ) = ad f = (xyz + si x, x z, x y). For which a is (ax l y, y + x /y) the gradiet of a fuctio i (some subset of) R? 6. Let be a curve i R give by r(t) = (cos t, si 3 t, t 4 ), where 0 t π. Evaluate (yz, xz, xy) dr.
Gree s Theorem. Let be the uit disk cetered at the origi i R. (a) Evaluate dx + x dy. (b) Evaluate arcta(e si x ) dx + y dy. (c) Evaluate (x 3 y 3 ) dx + (x 3 + y 3 ) dy. f f (d) Evaluate dx y x dy, give that f xx + f yy = 0.. Fid the area of the followig regios i R usig Gree s theorem. (a) The uit disk (b) The iverted cycloid: the regio bouded by the x axis ad the parametric curve x = a(t si t), y = a( cos t), 0 t π (c) The astroid: x /3 + y /3 a /3 (d) The ellipse: x /a + y /b 3. Let R be a regio i the plae with positively orieted. O, dr = (dx, dy). erive a expressio for ds i terms of dx ad dy. educe (usig the stadard Gree s theorem) the ormal form of Gree s theorem: F ds = F da. Notice that this is just a versio of the divergece theorem. 4. Let f be a smooth fuctio ad be a disk i R with outward uit ormal. For poits o, deote f/ to mea the directioal derivative of f i the directio of. Prove that f ds = f da.. Prove the idetity 6. Prove the idetity f f ds = P Q dx + P Q dy = R (f f + f ) da. [ ( P Q x P ) ( Q + P y x Q )] da. y
ivergece Theorem. I each of the followig situatios, evaluate (a) Let R be the uit ball cetered at the origi ad F = (x, y, 3z). F da. Assume is outward orieted. (b) Let R be the uit cube 0 x, y, z ad F = (y + si z, e si z +, xy + z). (c) Let R be the hemisphere x + y + z, z 0 ad F = (xz, yz, z ).. Let be the disk x + y, z =, orieted upward. Let be the coe x + y = z, 0 z, orieted dowward. Together, ad eclose a regio R. efie F = (x + e y, y + cos x, z). (a) Fid the flux of F across directly. (b) Itegrate F over R. (c) With o extra computatio, fid the flux of F across. geeralized? o you see how this problem could be 3. I each of the followig, use the method of the previous problem to fid the flux of F across. (a) Let be the hemisphere x + y + z = 9, z 0, outwardly orieted. Assume F = (x, 0, z). (b) Let be the coe z = 4 x + y, z 0, orieted upward. Assume F = (xy, yz, xz). 4. Let R be a solid regio with smooth boudary orieted outward. Assumig all fuctios are smooth, prove the followig idetities. (a) F da = 0 (b) f g da = (f g + f g) dv. R (c) (f g g f) da = (f g g f) dv. R (d) (x, y, z) da = 3 volume(r). tadard itegratio by parts is proved usig the product rule. But there are may differet product rules; each gives a differet itegratio by parts. (a) Let be a ball i R 3 with outward uit ormal vector. Assumig (fg) = f( G) + G f, prove that f( G) dv = fg da ( f) G dv. (b) Now let be the uit ball cetered at the origi. Evaluate e ( ) x +y +z (x, y, z) dv. (x + y + z ) 3/ You ca igore the sigularities at the origi (this could be made rigorous).
tokes Theorem. I each of the followig situatios, evaluate F da. (a) Let be the upper half (z 0) of the sphere x +y +z =, orieted upward, ad F = (x, xz, ye cos y ). (b) Let be the right half (x 0) of the sphere x +y +z =, orieted rightward, ad F = (x 3, y 3, 0). (c) Let be the part of the plae z = x with x +x+y 3, orieted upward, ad F = ((x+), 0, x ).. Let be the iteresectio of a (overtical) plae ad the cylider x + y = 4 i R 3. how that (x y) dx + (y + x) dy = 8π. 3. Let be a simple, closed, smooth curve o the sphere x + y + z =. how that ( xz, 0, y ) dr = 0. 4. Let be a smooth orieted surface with smooth boudary. Assumig all fuctios are smooth, prove the followig idetities. (a) f g dr = ( f g) da (b) (f g + g f) dr = 0. tadard itegratio by parts is proved usig the product rule. But there are may differet product rules; each gives a differet itegratio by parts. (a) Let be a smooth orieted surface with boudary. Give a smooth vector field G ad a smooth scalar fuctio f, show that f( G) da = ( f G) da + fg dr. (b) Now let be the coe z = x + y, 0 z, orieted dowward. efie G = ( y, x, arcta(xyz)e x ) ad evaluate z ( G) da (c) (Harder) Recall the idetity u (v w) = w (u v). Prove the vector equatio f da = fdr.
equeces ad eries. Fid the limit of the followig sequeces. (a) a = l / (b) a = ( /) 3 (c) a = + 3. Give the sequece (a ):, +, + +, + + +,... show that a + = a. Give that the limit of the sequece exists, use this formula to fid it. 3. Evaluate the followig series. (a) =0 3 3 (b) + si θ + si 4 θ + si 6 θ +, where 0 < θ < π/ (c) 4 + 8 7 + 6 37 + 87 + 4. Evaluate the followig series. (a) + 3 + 3 4 + + (b) + ( ) ( + ) (c) l ( + ) (d) (e) arcta( + ) arcta() =0 + + ( + ). A difficult series to evaluate is (a) + 4 + 6 + (b) + 3 + + 7 + (c) ( ) = π. Use this fact to evaluate the followig. 6
overgece of eries. etermie the covergece of the followig series. (a) (b) + (c) ( + (d) ( (e) si(/) (f) e 3 (g) arcta ) ) (h) + + (i) k / + k (j) [ ( + ) + ( ) ] +. etermie whether each series coverges absolutely, coditioally, or ot at all. (a) ( ) l (b) ( 4) (c) cos(π) 3. Use the error boud i the alteratig series test to approximate, withi decimal place accuracy, the followig values. The exact value of each series is give oly as trivia. (a) si = 3! +! 7! + (b) cos(/) = (/)! + (/)4 4! (/)6 6! + 4. uppose we wat to approximate l with digits of accuracy. If we use the alteratig series test, ( ) (a) How may terms of l = are eeded? (b) How may terms of l = ( ). For which real umbers p does the series coverge absolutely? oditioally? Not at all? are eeded? p p + 3 p 4 p +
eries: Miscellaeous. efie = + /3 + 3/3 + 4/3 3 + /3 4 +. (a) how that the series coverges. (b) Write out a series for 3. (c) ubstract the give equatio from the oe you just wrote. (d) Evaluate usig the previous part.. There is a costat γ, called the Euler-Mascheroi costat, so that k= k = l + γ + ɛ, where ɛ 0 whe. Use this fact to aswer the followig. (a) Use the above formula to show that (/) diverges. ( (b) Evaluate lim + + + + + ). 3. Fid costats A ad B so that 6 k (3 k+ k+ )(3 k k ) = Ak 3 k k + B k 3 k+ k+. Use this to evaluate 6 k (3 k+ k+ )(3 k k ). 4. The auchy odesatio test states that, give a positive oicreasig sequece a, a coverges if ad oly if a coverges. Use this test to check covergece of the followig: (a) / (b) /( log ) (c) /((log )(log log )). The Fiboacci umbers form a sequece F, where F 0 = F = ad F + = F + F + for all itegers. (a) Use telescopig to evaluate (b) It turs out that (amazigly) oes F coverge? 6. I 94, Ramauja proved that how that this series coverges. F F F +. ( F = 8 π = 980 + ) + ( k=0 (4k)!(03 + 6390k) (k!) 4 396 4k. ) +.