In many engineering and other applications, the. variable) will often depend on several other quantities (independent variables).

Transcription

1 II PARTIAL DIFFERENTIATION FUNCTIONS OF SEVERAL VARIABLES In man engineeing and othe applications, the behaviou o a cetain quantit dependent vaiable will oten depend on seveal othe quantities independent vaiables. Patial Dieentiation 1 Page

2 Eample 1 The volume V o a cone with base adius and height ht h, V 1 3 h Hee, V is the dependent vaiable that depends on the two independent vaiables and h. Eample The height h o a mountain above the gound level depends on the location o the spot on the gound level descibed b a plane Catesian coodinate sstem,, Height = h,. h is the dependent vaiable and it depends on the two independent d vaiables and, i.e., h is a unction o and. Patial Dieentiation Page

3 Eample 3 The speed s o a wate paticle in a steam depends, d in geneal, on its location descibed b Catesian coodinates,,, and time t, speed = s,,, t The dependent speed s is a unction o ou independent vaiables,, and t. Patial Dieentiation Page 3

4 Domain The domain o a unction is the egion o values o the independent vaiable such that the unction is deined. Eample 4 Conside a datboad with unit adius and the distance o a landing dat om the cente o the boad is measued, we have, 1. The domain o this unction is the above closed unit disc. The unction is deined onl within this domain. Patial Dieentiation Page 4

5 Gaphs The gaph o a unction o two vaiables, =, is a suaces in space. Fo each point, on the plane, mak the point P,, o P,,. Do this o all, values in the domain., P,,,, Patial Dieentiation Page 5

6 Eample 5 Conside the unction, 1 Since we equie to be eal, so 1 1 The domain o the unction is a close disc with adius 1 cented at the oigin Patial Dieentiation Page 6

7 LIMIT OF A FUNCTION The limit o a unction, as, o, o is the numbe L that the unction value appoaches as the point, is abita close to but not equal to the point o, o, and we wite lim,,, L The limit o a unction o thee o moe vaiables is deined in a simila manne. Patial Dieentiation Page 7

8 Eample 6 Conside the unction, at the oigin,. Along the -ais,, =, = ; along the line =,, = 1; and along the -ais,, =, and is undeined. Thus the limit o the unction does not eist at the oigin. Patial Dieentiation Page 8

9 Eample 7 lim, 5, Eample lim 1 11, 1, 1 Patial Dieentiation Page 9

10 Popeties o limits lim I, L1 &,, lim,, g, then [, g, ] L1 L lim,, L lim,,, g, L 1 L lim,, k, k L 1 lim, L 1,, g, L Patial Dieentiation Page 1

11 Continuit A unction, is continuous at the point o, o i 1 is deined at o, o, i.e., the unction has a unique eal value at o, o ;, eists; and lim,, 3 lim,,,,. I a unction is continuous at eve point in its domain, then it is a continuous unction. Patial Dieentiation Page 11

12 Eample 9 The unction, = + 3 is a continuous unction. Eample 1 The unction, is continuous at eve point ecept, at which it is undeined. What s the domain o this unction? Eample 11 is a continuous unction.,, 1 Patial Dieentiation Page 1

13 Popeties o continuous unctions 1 The sums, dieences and poducts o continuous unctions ae continuous. In addition, the quotient o two continuous unctions is continuous povided it is deined. Polnomial unctions ae alwas continuous. Rational unctions ae continuous at eve point the ae deined. 3 I =, and w = g ae continuous, then the composite unction w = g, is continuous. In geneal, the composite o continuous unctions is continuous. Patial Dieentiation Page 13

14 Eample 1 The unctions and e ae continuous, so the composite unction e is continuous. Eample 13 The unctions and w cos ae continuous, so the 1 unction w cos 1 is continuous. Eample 14 The unctions 1 and ln ae continuous, so the unction ln1 is continuous. Patial Dieentiation Page 14

15 PARTIAL DERIVATIVE Patial deivative o with espect to, Vetical ais o, o, o, o =, Tangent o Gaph o =, o in the plane = o O o, o o +, o Hoiontal ais o Patial Dieentiation Page 15

16 The cuve =, o is a one-dimensional cuve, i.e., it is a unction o one vaiable since is held constant t at o. Its deivative with espect to at = o is d d,, o lim This limit is called the patial deivative o, with espect to at the point o, o., Geometicall, this patial deivative is the slope o the tangent to the cuve, o. This deivative is called patial since is held constant when the limit is evaluated. Patial Dieentiation Page 16

17 Notations o the patial deivative: 1 o o o, o, 3 o, Eample 15 Find o the unction. d d d d Patial Dieentiation Page 17

18 Eample 16 lnl Find o the unction. Eample 17 e ln e ln e ln Find o the unction. e sin d e sin sin e e sin d e sin cos Patial Dieentiation Page 18

19 Patial deivative o with espect to, o, o, o, o Vetical ais Tangent =, Gaph o = o, in the plane = o o, o o, o + Hoiontal ais Patial Dieentiation Page 19

20 The cuve = o, is a one-dimensional cuve, i.e., it is a unction o the vaiable since is held constant t at o. Its deivative with espect to at = o is d d o o, lim o,, This limit is the patial deivative o, with espect to at the point o, o. Geometicall, this patial deivative is the slope o the tangent to the cuve o,. This deivative is onl patial since is held constant when the limit is evaluated. Patial Dieentiation Page

21 Notations o the patial deivative: 1 o, o, o o o o o 3, Eample 18 Find o the unction. d d Patial Dieentiation Page 1

22 Eample 19 Find ln o the unction e. 1 Eample Find o the unction e sin. e sin e sin e cos Patial Dieentiation Page

23 Eample 1 cos Find the patial deivatives o the unction with espect to each vaiable. cos,, cos d d cos d cos Patial Dieentiation 3 Page

24 d cos d d sin 1 cos 1 cos cos cos Patial Dieentiation 4 Page

25 Eample Accoding to the cuent phsical theo, an bod nea a black hole one o the possible inal states o stas will epeience a tansvese oce known as the tidal oce pulling the bod apat. Conside this tidal oce F acting between ou head and abdomen tip when ou obit ound a black hole, it is given k s m, F c whee m is the black hole s mass, c is the cicumeence o ou obit, s is the distance between ou head to the end o ou abdomen, and k is a constant. Fo a given black hole, detemine the change o the tidal oce with espect to the cicumeence o ou obit assuming that ou bod is made o ve stong mateials so that ou bod would not be lengthened b the oce. Patial Dieentiation Page 5 3

26 HIGHER ORDER DERIVATIVES Fo a unction o two vaiables,, with dieentiable patial deivatives, thee ae ou second patial deivatives: 1 o ; o ; 3 o ; 4 o. Patial Dieentiation Page 6

27 Eample 3 de d d d cos e 5 cos 3, e d de d d d d d d 5 sin 3 5 cos 3 e d 3 5 cos 3 d e e 5 cos 5 sin 3 e e 5 cos 5 sin sin 3 e Patial Dieentiation 7 Page

28 Eample 4 The two-dimensional Laplace equation is given b Show that the unction Laplace equation., tan 1 satisies the Patial Dieentiation Page 8

29 Mied deivative theoem Fo a unction i ae deined and,,,, continuous, then we have, i.e., the ode o dieentiation is immateial. The theoem usuall holds o unctions that we encounte in engineeing poblems in engineeing poblems. Similal o highe ode deivatives the ode o Similal, o highe ode deivatives, the ode o dieentiation is immateial povided all the patial deivatives ae deined and continuous. Patial Dieentiation 9 Page

30 Eample 5 5 Show that 3 is eo whee, e sin cos. The patial deivatives ae continuous, the ode o dieentiation is not impotant t Patial Dieentiation Page 3

31 Eample 6 Fo the unction, vei that 3 3,, 3 3 Patial Dieentiation 31 Page

32 CHAIN RULE Function o one vaiable: is a unction o and is a unction o, d d d d d d d d d d Patial Dieentiation Page 3

33 Functions educible to a unction o one vaiable, and that and ae both unctions o anothe vaiable t, ie i.e. t and t : d d d d dt dt dt dt d dt, d dt t Patial Dieentiation Page 33

34 Eample 7 d I, ind with the help o the chain ule. and, sin t t e dt d e d d d de e d d d d e ; sin sin ; cos sin t dt dt dt d dt dt dt d 1 d d d Since dt d dt d dt d t t e t t t e t e e t t sin cos * sin *1 cos t t t t e t t sin cos 1 Patial Dieentiation 34 Page

35 Eample 8 d v t Given that u, u t 1 and v e, ind. dt Patial Dieentiation Page 35

36 Functions o thee o moe vaiables w,, ; t; t; t gives dw d d d dt dt dt dt, then the chain ule d dt d dt w d dt t d dt Patial Dieentiation Page 36

37 Eample 9 dw 3 Find i w ; ln t; t ; sin t. dt Patial Dieentiation Page 37

38 Case when some o the independent vaiables ae also unctions o the emaining i independent d vaiable: v v,, t; t; t dv dt v t v d dt v d dt v d dt v v d dt t v v t t dt dt 1 Patial Dieentiation Page 38

39 Functions educible to a unction o two vaiables., ;, ;, ;,, v u v u v u w The chain ule will assume the om w w w w ; u w u w u w u w. v w v w v w v w Patial Dieentiation 39 Page

40 u w vv u w w w u v w w u v v Patial Dieentiation Page 4

41 Eample 3 s I w ; e cos s; e sin s; e, ind b the chain ule. w and w s Patial Dieentiation Page 41

42 Case :, and w w ; d dw w. ; d dw w d d w d dw w Patial Dieentiation 4 Page

43 Eample 31 uv Find and i e ; u v; e. u v Patial Dieentiation Page 43

44 Eample 3 I a b, whee a and b ae constants, show that b a. Hint: Let u a b, then u and u is a unction o and. Patial Dieentiation Page 44

45 Remaks on the Chain Rule In man phsical and engineeing poblems, the quantities o vaiables ae inte-dependent and inding patial deivatives can pove to be tick at times. The most impotant t thing is to state t cleal l which h quantities o vaiables ae taken as independent vaiables. Patial Dieentiation Page 45

46 IMPLICIT FUNCTIONS I the unctions ae not given eplicitl, we have to ascetain which o the vaiables ae independent vaiables. Ate that, we can dieentiate the implicit unction on both sides to obtain the equied patial deivatives. Patial Dieentiation Page 46

47 Eample 33 Conside an implicit unction,,. Assume that it detemines as a unction o and, and that, show that. Note that in evaluating all the patial deivatives o taken as a unction o the thee vaiables,,., it is Patial Dieentiation Page 47

48 Eample 34 I and,, then show that 1 u u u u, u cos sin 1 u u u u Patial Dieentiation 48 Page

49 Eample 35 u and v ae unctions o and deined though the ollowing elations ; 6 4,,, uv v u 9,,, v u v u g Find,, and. g u u v v Do not detemine the geneal omulae o such implicit unctions instead dieentiate the elations and solve unctions, instead dieentiate the elations and solve o the equied esults Patial Dieentiation 49 Page

50 THEOREM ON LINEAR APPROXIMATION The unction, changes b an amount in going om, to, :,,. Fo a ied point,, this is a unction o and such that when. Then the linea appoimation is whee the patial deivatives ae evaluated at the ied point,. Patial Dieentiation Page 5

51 Eample 36 V h is detemined om measuements o and h that ae in eo b 1%, what is the appoimated pecentage eo in V? Patial Dieentiation Page 51

52 Eample 37 The aea o a tiangle ABC is given b S = ½ab sin C. In suveing a paticula tiangula plot o land, a an b ae measued to be 5m and 67m espectivel, and C is ead to be 6 o. B how much appoimatel is the computed aea in eo i a and b ae in eo b.15m each and C is in eo b o? Patial Dieentiation Page 5

53 TOTAL DIFFERENTIAL Fo a unction, that has continuous ist ode patial deivatives, the total dieential d o at the point, is d d d As and ae independent d vaiables, we have d and d. So d Patial Dieentiation Page 53

54 Eample 38 At a cetain instant the adius o a closed ight cicula clinde is 6m and is inceasing at the ate. ms -1, while the altitude is 8m and is deceasing at the ate.4 ms -1. Find the time ate o change a o the volume and b o the suace at that instant. Eample 39 Find and i e ; s st; st t. s t The dieentials ae Patial Dieentiation Page 54

55 MAXIMUM & MINIMUM VALUES Functions o two vaiables, like unctions o one vaiable, can have elative o local and absolute maimum and minimum values. Relative maimum elative minimum i occus when the unction value at that point is lage smalle than the unction values o its suounding points. Absolute maimum minimum point is the point at which the unction assumes its maimum minimum value ove its whole domain. Patial Dieentiation Page 55

56 Eample 4, This unction has an absolute minimum at the point,,. Patial Dieentiation Page 56

57 Eample 41, = The actional pat o the actional pat o o, This unction has ininite numbe o elative maima and has the minimum value eo wheneve o is a non-negative intege. Patial Dieentiation Page 57

58 Eample 4, sin *sin * e This unction has ininite numbe o elative maima and elative minima. Patial Dieentiation Page 58

59 Fo maimum and minimum points that do not lie on the bounda o the domain, we obseve that eithe 1 the tangent planes ae hoiontal so that ; o o both o them ails to eist at these points. Necessa but not suicient condition o elative maimum/minimum: ; o one o both o them ails to eist. Patial Dieentiation Page 59

60 Saddle Point is onl a necessa condition o elative maimum/minimum but not suicient. Thee ae points that satis this condition but ail to be eteme points. Patial Dieentiation Page 6

61 Eample 43, At the point,,, but the unction is at maimum along cetain diections and at minimum along othe diections, so it is not an eteme point. Such points ae saddle points. Patial Dieentiation Page 61

62 Test o elative maimum / minimum 1. Check o citical points: a I one o both o and ails to eist, then inspect the unction aound that point to detemine whethe it is a maimum, a minimum o neithe one o them. b I, and i the ist and second ode patial deivatives ae continuous, poceed to step and use the second deivative test. O couse, ou ma inspect the unction diectl aound the point to ascetain its natue. Patial Dieentiation Page 6

63 Second deivative test. I a, b a, b, detemine this is sometimes known as the Hessian at a, b : a I o equivalentl, and at, a, b then has a elative maimum at a, b. b I o equivalentl, and at a, b, then has a elative minimum at a, b. c I at a, b, then has a saddle point at a, b. d I, the test is inconclusive at a, b. We have to seek othe means to detemine the behaviou o at a, b. Patial Dieentiation Page 63

64 Eample 44 3 Test the suace, o maima, minima, and saddle points. Find the unction values at these points The domain o, has no bounda points, and all its deivatives ae polnomial unctions o,, thus ae continuous. So maimum/minimum can onl occu at the citical points. Patial Dieentiation Page 64

65 Eample 45 Find the absolute maima and minima i o the unction, 1 which is deined ove the closed tiangula plate in the ist quadant bounded b the lines =, = 4, =. The domain o the unction is the tiangle OAB A B O Conside points on the bounda o the tiangle and citical points inside the tiangle. Patial Dieentiation Page 65

66 Eample 45Cont d Conside points on the bounda o the tiangle and citical points inside the tiangle Patial Dieentiation Page 66

67 Eample 46 The method o least squaes An electical cuent is ed into an electonic sstem and the output voltage is measued. The epeiment is caied out n times and a set o data 1, 1,,,, n, n is obtained, whee i s ae the input cuents I s ae the output voltage. Assuming a linea elationship between the input cuent and the output voltage, we would like to it the best staight line o the set o data. Patial Dieentiation Page 67

68 Eample 46 Cont d Let i be the pedicted voltage at i accoding to the itted staight line. Then the eo at i is i i and the method o least squaes equie that the itted line should minimie the squae o the total sum o eos S in i1 i i Let the equation o the itted line to be = a + b, then S i n i1 a b i i Poblem: Detemine a and b such that S is a minimum. Patial Dieentiation Page 68

69 SUMMARY ON MAXIMUM / MINIMUM VALUES 1 Relative maimum/minimum values inteio points Step 1 Detemine the citical points: a, o both o them ails to eist; o b. I a then investigate the behaviou o the unction nea the citical points manuall such as sketching some points o the suace. I b go to step. Patial Dieentiation Page 69

70 Step Detemine the disciminant o hessian o the unction, and conduct the second deivative test : = < > The test is Saddle point a o : inconclusive. Minimum point. Find othe was to detemine the b o : natue o the citical Maimum point. point. Patial Dieentiation Page 7

71 Absolute maimum/minimum values Detemine the maimum/minimum values o the unction at its bounda, and then ind the oveall maimum/minimum values among these bounda points and the elative maimum/minimum points ound in 1. Patial Dieentiation Page 71

72 APPENDIX I Function Deivative n n n-1 e e ln 1/ a a lna sin cos cos -sin tan sec Patial Dieentiation Page 7

73 Function sinh cosh tanh Deivative cosh sinh sech sin cos tan Patial Dieentiation Page 73