Polytechnic Institute of NYU MA 2122 Worksheet 4

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1 Polytechnic Institute of NYU MA 222 Worksheet 4 Print Nme: ignture: ID #: Instructor/ection: / Directions: how ll your work for every problem. Problem Possible Points Totl 00

2 MA222 Fll 20 Worksheet 04, Your signture: () Accoring to Coulomb s Lw, the electrosttic fiel E t the point r ue to chrge q t the origin is given by () ketch E when q =. (b) Fin iv E. E( r) = q r r 3 (c) Let be the sphere of rius centere t the origin n oriente outwr. how tht the flux of E through is 4πq. () Coul you hve use the Divergence Theorem in prt (c)? Explin why or why not. (e) Let be n rbitrry, close, outwr-oriente surfce surrouning the origin. how tht the flux of E through is gin 4πq. [Hint: Apply the Divergence Theorem to the soli region lying between smll sphere n the surfce.]

3 MA222 Fll 20 Worksheet 04, Your signture: (2) () Fin the flux of F = x 3 î + y 3 ĵ + z 3ˆk through the close surfce bouning the soli region x 2 + y 2 4, 0 z 5, oriente outwr. (b) The region W lies between the spheres x 2 +y 2 +z 2 = 4 n x 2 +y 2 +z 2 = 9 n within the cone z = x 2 + y 2 with z 0; its bounry is the close surfce,, oriente outwr. Fin the flux of F = x(y 2 +z 2 )î+y(x 2 +z 2 )ĵ +z(x 2 +y 2 )ˆk out of.

4 MA222 Fll 20 Worksheet 04, Your signture: (3) how ech of the following: () If g(x, y, z) is sclr vlue function n F (x, y, z) is vector fiel, then iv(g F ) = (gr g) F + g iv F. (b) If G(x, y, z) is ny vector fiel whose components hve continuous secon prtil erivtives, then iv curl G = 0. (c) If f(x, y, z) hs continuous secon prtil erivtives, then curl gr f = 0.

5 MA222 Fll 20 Worksheet 04, Your signture: (4) Fin the flux of the vector fiel F through the prmeterize surfce. () F = zî + xĵ n is oriente upwr n given, for 0 s, t 3, by x = s 2, y = 2s + t 2, z = 5t.

6 MA222 Fll 20 Worksheet 04, Your signture: (b) F = x 2 y 2 zˆk n is the cone x 2 + y 2 = z, with 0 z R, oriente ownwr. (Hint: Prmeterize the cone using cylinricl coorintes.)(ee Figure ) Figure

7 MA222 Fll 20 Worksheet 04, Your signture: (5) Use tokes Theorem to compute the integrl F (x, y, z) = yzî + zxĵ + xyˆk curl F A where n is the prt of the sphere x 2 +y 2 +z 2 = 4 tht lies insie the cyliner x 2 +y 2 = n bove the xy-plne.

8 MA222 Fll 20 Worksheet 04, Your signture: (6) For constnt p, consier the vector fiel E = r r p. () Fin curl E. (b) Fin the omin of E. (c) For which vlues of p oes E stisfy the curl test? For those vlues of p, fin potentil function for E.

9 MA222 Fll 20 Worksheet 04, Your signture: (7) Use tokes Theorem to evlute C F r. () F (x, y, z) = x 2 zî + xy 2 ĵ + z 2ˆk n C is the curve of intersection of the plne x + y + z = n the cyliner x 2 + y 2 = 9 oriente counterclockwise s viewe from bove. (b) F (x, y, z) = (y z)î + (x + z)ĵ + xyˆk n C is circle of rius 3 centere t (2,, 0) in the xy-plne, oriente counterclockwise when viewe from bove. Is F pth-inepenent?

10 MA222 Fll 20 Worksheet 04, Your signture: Multivrible Integrl The efinite integrl of f, continuous function of two vribles, over R, the rectngle x b, c y, is clle ouble integrl, n is limit of Riemnn sums f A = lim f(u ij, v ij ) x y. R x, y 0 i,j A triple integrl of f, continuous function of three vribles, over W, the box x b, c y, p z q in 3-spce, is efine in similr wy using three-vrible Riemnn sums. Interprettions If f(x, y) is positive, f A is the volume uner grph of f bove the region R R. If f(x, y) = for ll x n y, then the re of R is A = A. If R R f(x, y) is ensity, then f A is the totl quntity in the region R. The R verge vlue of f(x, y) on the region R is f A. Are of R R Iterte integrls Double n triple integrls cn be written s iterte integrls b f A = f(x, y) xy R W f V = c q b p c f(x, y, z) xyz. Other orers of integrtion re possible. For iterte integrls over nonrectngulr regions, limits on outer integrl re constnts n limits on inner integrls involve only the vribles in the integrls further out. Integrls in other coorinte systems When computing ouble integrls in polr coorintes, put A = r r θ or A = r θ r. Cylinricl coorintes re given by x = r cos θ, y = r sin θ, z = z, for 0 r <, 0 θ 2π, < z <. phericl coorintes re given by x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ, for 0 ρ <, 0 φ π, 0 θ 2π. When computing triple integrls in cylinricl or sphericl coorintes, put V = r r θ z for cylinricl coorintes, V = ρ 2 sin φ ρ φ θ for sphericl coorintes. Other orers of integrtion re lso possible. For chnge of vribles x = x(s, t), y = y(s, t), the Jcobin is (x, y) (s, t) = x s y t x t y x x s = s t y y s t

11 MA222 Fll 20 Worksheet 04, Your signture: Line Integrls The line integrl of vector fiel F long n oriente curve C is F n, r = lim F n ( r i ), r i = lim F ( r i ) r i, C r i 0 i=0 r i 0 i=0 where the irection of Dt r i is the irection of the orienttion. The line integrl mesures the extent to which C is going with F or ginst it. For oriente curves C, C n C 2, F, r = F, r, where C C C is the curve C prmeterize in the opposite irection, n F, r = C +C 2 F, r + F, r, where C + C 2 is the curve obtine by joining the C C 2 enpoint of C to the strting point of C 2. The work one by force F long curve C is F, r. The circultion C of F roun n oriente close curve is F, r. Given prmeteriztion of C, r(t), for t b, the line integrl cn be clculte s b F, r = F ( r(t)), r (t) t. C Funmentl Theorem for Line Integrls (FTL or FTLI): uppose C is piecewise smooth oriente pth with strting point P n enpoint Q. If f is function whose grient is continuous on the pth C, then gr f, r = f(q) f(p ). C Pth-inepenent fiels n grient fiels A vector fiel F is si to be pth-inepenent, or conservtive, if for ny two points P n Q, the line integrl F C, r hs the sme vlue long ny piecewise smooth pth C from P to Q lying in the omin of F. A grient fiel is vector fiel of the form F = gr f for some sclr function f, n f is clle potentil function for the vector fiel F. A vector fiel F is pth-inepenent if n only if F is grient vector fiel. A vector fiel F is pth-inepenent if n only if F, r = 0 for every close curve C. If F C C is grient fiel, then F 2 x F y = 0. The quntity F 2 x F is clle the y 2-imension or sclr curl of F.

12 MA222 Fll 20 Worksheet 04, Your signture: Green s Theorem: uppose C is piecewise smooth simple close curve tht is the bounry of n open region R in the plne n oriente so [ tht ] the region is on the left s we move roun the curve. uppose F F = = F F i + F 2 j is smooth 2 vector fiel efine t every point of the region R n bounry C. Then ( F F2, r = x F ) xy. y C Curl test for vector fiels in 2-spce: R If F 2 x F y = 0 n the omin of F hs no holes, then F is pth-inepenent, n hence grient fiel. The conition tht the omin hve no holes is importnt. It is not lwys true tht if the sclr curl of F is zero then F is grient fiel. urfce Integrls A surfce is oriente if unit norml vector n hs been chosen t every point on it in continuous wy. For close surfce, we usully choose the out wr orienttion. The re vector of flt, oriente surfce is vector A whose mgnitue is the re of the surfce, n whose irection is the irection of the orienttion vector ˆn. If v is the velocity vector of constnt flui flow n A is the re vector of flt surfce, then the totl flow through the surfce in units of volume per unit time is clle the flux of v through the surfce n is given by v, A. The surfce integrl or flux integrl of the vector fiel F through the oriente surfce is F, A = lim F, A. A 0 where the irection of A is the irection of the orienttion. If v is vrible vector fiel then v, A is the flux through the surfce. imple flux integrls cn be clculte by putting A = ˆnA n using geometry or converting to ouble integrl. The flux through grph of z = f(x, y) bove region R in the xy-plne, oriente upwr, is F, A = F (x, y, f(x, y)) F 2 (x, y, f(x, y)), f x f y xy R F 3 (x, y, f(x, y)) = F (x, y, f(x, y)) F 2 (x, y, f(x, y)), f x f R F 3 (x, y, f(x, y)) + f 2 x + fy 2 y + fx 2 + fy 2 xy

13 MA222 Fll 20 Worksheet 04, Your signture: The flux through cylinricl surfce of rius R n oriente wy from the z-xis is F, A = F (R, θ, z) F 2 (R, θ, z), x y R zθ T F 3 (R, θ, z) R 0 = F F 2, cos θ sin θ R zθ T 0 F 3 where T is the θz-region corresponing to. The flux through sphericl surfce of rius R n oriente wy from the origin is F, A = F (R, φ, θ) F 2 (R, φ, θ), x y R φ sin φrθ T F 3 (R, φ, θ) R z = F sin φ cos θ F 2, sin φ sin θ R 2 sin φ φθ T cos φ F 3 where T is the θφ-region corresponing to. The flux through prmeterize surfce, prmeterize by r = r(s, t), where (s, t) vries in prmeter region R, is F, A = F (s, t) F 2 (s, t), r R F 3 (s, t) s r st t ( = F r F 2 s r ) t r, R F 3 r s r s r t st t where F = F ( r(s, t)) = F (s, t) F 2 (s, t). We choose the prmeteriztion so tht F 3 (s, t) r s r is never zero n points in the irection of ˆn everywhere. t The Are of Prmeterize urfce The re of surfce which is prmeterize by r = r(s, t), where (s, t) vries in prmeter region R, is given by A = r R s r t st. }{{} A on the surfce

14 MA222 Fll 20 Worksheet 04, Your signture: Divergence n Curl Divergence Definition of Divergence Geometric efinition: The ivergence of F is F, A iv F (x, y, z) = lim Volume 0 Volume of Here is sphere centere t (x, y, z), oriente outwrs, tht contrcts own to (x, y, z) in the limit. Crtesin coorinte efinition: If F = F = F i + F 2 j + F 3 k, then iv F =, F = / x / y, F F 2 = F / z x + F 2 y + F 3 z F 3 The ivergence cn be thought of the outflow per unit volume of the vector fiel. A vector fiel F is si to be ivergence free or solenoil if iv F = 0 everywhere tht F is efine. Mgnetic fiels re ivergence free. The Divergence Theorem If W is soli region whose bounry is piecewise smooth surfce, n if F is smooth vector fiel which is efine everywhere in W n on, then F, A = iv F V, W where is given the outwr orienttion. In wors, the Divergence Theorem sys tht the totl flux out of close surfce is the integrl of the flux ensity over the volume it encloses. Curl The circultion ensity of smooth vector fiel F t (x, y, z) roun the irection of the unit vector n is efine to be Circultion roun C circˆn F (x, y, z) = lim Are 0 Are insie C F, r = lim Are 0 Are insie C Circultion ensity is clculte using the right-hn rule. C F 2 F 3

15 MA222 Fll 20 Worksheet 04, Your signture: Definition of curl Geometric efinition The curl of F, written curl F, is the vector fiel with the following properties () The irection of curl F (x, y, z) is the irection ˆn for which circˆn F (x, y, z) is gretest. (b) The mgnitue of curl F (x, y, z) is the circultion ensity of F roun tht irection. Compre these circˆn F = curl F, ˆn fû = gr f, û Crtesin coorinte efinition: If F = F = F i + F 2 j + F 3 k, then curl F = F i j k = / x / y / z F F 2 F 3 Curl n circultion ensity re relte by circˆn F = curl F, ˆn. A vector fiel is si to be curl free or irrottionl if curl F = 0 everywhere tht F is efine. Given n oriente surfce with bounry curve C we use the right-hn rule to etermine the orienttion of C. tokes Theorem If is smooth oriente surfce with piecewise smooth, oriente bounry C, n if F is smooth vector fiel which is efine on n C, then F, r = curl F, A. C tokes Theorem sys tht the totl circultion roun C is the integrl over of the circultion ensity. A curl fiel is vector fiel F tht cn be written s F = curl G for some vector fiel G, clle vector potentil for F. Reltion between ivergence, grient, n curl The curl n grient re relte by curl gr f = 0. Divergence n curl re relte by iv curl G = 0. The curl test for vector fiels in 3-spce uppose tht curl F = 0, n tht the omin of F hs the property tht every close curve in it cn be contrcte to point in smooth wy, stying t ll times within the omin. Then F is pth-inepenent, so F is grient fiel n hs potentil function. The ivergence test for vector fiels in 3-spce uppose tht iv F = 0, n tht the omin of F hs the property tht every close surfce in it is the bounry of soli region completely contine in the omin. Then F is curl fiel. F 2 F 3

16 MA222 Fll 20 Worksheet 04, Your signture: Geometry Formuls Here V is the volume, is the surfce re, h is the height n r is the rius. Cyliner: V = πr 2 h, = 2πrh Cone: V = 3 πr2 h phere: V = 4 3 πr3, = 4πr 2 Physics formuls: The ccelertion ue to grvity, g: g = 9.8m/sec 2, or g = 32ft/sec 2. Mss ensity of wter = 000 kg/m 3, Weight ensity of wter = 62.4 lbs/ft 3. Force = mss ccelertion Work = Force istnce The center of mss, x, of n object lying on the x-xis between x = n b xδ(x) x x = b, with mss ensity δ(x) is given by x = totl mss b Arc length of curve y = f(x) from x = to x = b: L = + (f (x)) 2 x Integrtion by Prts: uv = uv vu or Numericl Approximtions: uv x = uv vu x TRAP(n) = LEFT(n)+RIGHT(n) ; IMP(n) = 2MID(n)+TRAP(n) 2 3 Useful Integrls for Comprison: x converges for p > n iverges for p. xp x converges for p < n iverges for p. xp 0 0 e x x converges for > 0. ummry of solutions to y + by + cy = 0. If b 2 4c > 0, then r n r 2 re two istinct solutions of the chrcteristic eqution n y = C e r t + C 2 e r 2t, where C n C 2 re constnts. If b 2 4c = 0, then there is only one solution of the chrcteristic eqution, r = b/2, n y = C te rt + C 2 e rt. If b 2 4c < 0, then the solutions of the chrcteristic eqution re of the form r = α ± βj n y = C e αt cos(βt) + C 2 e αt sin(βt).

17 MA222 Fll 20 Worksheet 04, Your signture: nth egree Tylor Polynomil of f(x) centere t x = : f(x) = f() + f ()(x ) + f () (x ) 2 + f () (x ) f (n) () (x ) n 2! 3! n! Tylor series of f(x) centere t x = : f(x) = f() + f ()(x ) + f () (x ) 2 + f () (x ) 3 + 2! 3! Tylor eries of importnt functions: sin(x) = x x3 3! + x5 5! x7 7! + cos(x) = x2 2! + x4 4! x6 6! + e x = + x + x2 2! + x3 3! + x4 4! + x = + x + x2 + x 3 + for < x < ln( + x) = x x2 2 + x3 3 x4 4 + for < x ( + x) p p(p ) = + px + x 2 p(p )(p 2) + x 3 + for < x < 2! 3! Finite Geometric eries: Infinite Geometric eries: + x + x x n = ( xn ) x + x + x 2 + = Rtio Test: For the series n, suppose, x n+ lim = L. n n If L <, then the series converges. If L >, then the series iverges. If L =, then the test fils. for x <

18 MA222 Fll 20 Worksheet 04, Your signture: Differentition formuls x (xn ) = nx n x (ln x ) = x x (rcsin(x)) = x 2 x (ex ) = e x (sin(x)) = cos x x x (tn(x)) = sec2 x (sec(x)) = sec x tn x x x (rccos(x)) = x 2 Here, b, c, re constnts. A hort Tble of Inefinite Integrls I. Bsic Functions. x n x = n + xn+ + C, (n ) x = ln x + C 6. x 3. x x = ln x + C ln x x = x ln x x + C x (x ) = (ln ) x (cos(x)) = sin x x x (cot(x)) = csc2 x (csc(x)) = csc x cot x x x (rctn(x)) = + x 2 sin x x = cos x + C cos x x = sin x + C tn x x = ln cos x + C II. Proucts of e x, cos x, n sin x 8. e x sin(bx) x = 2 + b 2 ex [ sin(bx) b cos(bx)] + C 9. e x cos(bx) x = 2 + b 2 ex [ cos(bx) + b sin(bx)] + C 0. sin(x) sin(bx) x = [ cos(x) sin(bx) b sin(x) cos(bx)] + C, b 2 2 b. cos(x) cos(bx) x = [b cos(x) sin(bx) sin(x) cos(bx)] + C, b 2 2 b 2. sin(x) cos(bx) x = [b sin(x) sin(bx) + cos(x) cos(bx)] + C, b 2 2 b

19 MA222 Fll 20 Worksheet 04, Your signture: III. Prouct of Polynomil p(x) with ln x,e x, cos x, n sin x 3. x n ln x x = n + xn+ ln x (n + ) 2 xn+ + C, n, x > 0 4. p(x)e x x = p(x)ex 2 p (x)e x + 3 p (x)e x + C ( ) (signs lternte) 5. p(x) sin x x = p(x) cos(x) + 2 p (x) sin(x) + 3 p (x) cos(x) + C ( ) (signs lternte in pirs) 6. p(x) cos x x = p(x) sin(x) + 2 p (x) cos(x) 3 p (x) sin(x) + C ( ) (signs lternte in pirs) IV. Integer Powers of sin x n cos x 7. sin n x x = n sinn x cos x + n sin n 2 x x, n positive n 8. cos n x x = n cosn x sin x + n cos n 2 x x, n positive n 9. sin m x x = cos x m sin m x + m 2 m sin m 2 x x, m, m positive 20. sin x x = 2 ln cos x cos x + + C 2. cos m x x = sin x m cos m x + m 2 m cos m 2 x x, m, m positive 22. cos x x = 2 ln sin x + sin x + C 23. sin m x cos n x x : If n is o, let w = sin x. If both m n n re even n non-negtive, convert ll to sin x or ll to cos x (using sin 2 x + cos 2 x = ), n use IV-7 or IV-8. If m n n re even n one of them is negtive, convert to whichever function is in the enomintor n use IV-9 or IV-2. The cse in which both m n n re even n negtive is omitte. V. Qurtic in the Denomintor 24. x 2 + x = ( x ) 2 rctn + C, 0 bx + c 25. x 2 + x = b 2 2 ln x c ( x ) rctn + C, 0

20 MA222 Fll 20 Worksheet 04, Your signture: 26. (x )(x b) x = (ln x ln x b ) + C, b ( b) cx (x )(x b) x = [(c + ) ln x (bc + ) ln x b ] + C, ( b) b VI. Integrns involving 2 + x 2, 2 x 2, x 2 2, > 0 x ( x ) x = rcsin + C 2 x 29. x2 ± = ln x + x 2 ± 2 + C ± x 2 x = (x ) 2 2 ± x ± x x + C 2 x x = (x ) x x2 x + C 2