11.6 Directional Derivatives and the Gradient Vector

Transcription

1 6 Diectional Deivatives and the Gadient Vecto So a we ve ound the ate o change o a unction o two o moe vaiables in the diection paallel to the ais (, we set constant, = b plane), and in the diection paallel to the ais (, we set constant, = a plane) z T Suace, z = (, ) T What i we want to ind the ate o change o a unction (o two o moe vaiables) in an diection? all it in the diection o the unit vecto, u z 3 T Suace, z = (, ) T u In the diection o the unit vecto, u

2 TE 6A Show them TE 6A animate and visuall It opens up showing the slope o the ed line Diectional Deivative we e inding a deivative in the diection we want Slope o a tangent line How can we deine it in this new diection, u? Remembe, eve point in a plane can be deined as a scala multiple o a and b u = a, b a+ b o witten in unction notation ( a, ) + ( b, ) is a deivative in the diection o u (an unit vecto) and has notation Du (, ) = (, ) + (, ) Du a b I is a dieentiable unction o and, then has a diectional deivative in the diection o an unit vecto u = a, b So we e using these patials and o witten and to deine an diection Do ou see ou could wite it as: ( ) ( ) ( ) D, =,,, a, b D (, ) is the ate o change o the u Du (, ) = (, ), (, ) u u unction in the diection o unit vecto u This ist vecto in the dot poduct occus so oten, the give it a special name, the gadient o, and use notation (gad ), o moe commonl, ponounced del Deinition: I is a unction o two vaiables and, then the gadient o is the vecto unction, =,,, ( ) ( ) ( ) and I the diection is given as an angle, θ, then its diection is u = a, b = u cos θ, u sinθ u is a unit vecto so u = a, b = cos θ, sinθ θ

3 Eample : 3 Find the diectional deivative o the unction (, ) = 3+ 4 at the point ( ) the unit vecto given b θ = π 6, in the diection o I, o eample, (, ) (, ) sailing in the diection o hoizontal mete epesented the depth o the ocean in metes You ae on a boat positioned at θ = π, the depth o the ocean is inceasing 39 vetical metes pe 6 This visual is pictued in E, page 79-79, Figue 5 Look at the cuve whee z o (, ) intesects the plane at (,,0 ) The diectional deivative is the slope o the tangent line to this cuve in the diection o u It is the ate o change o z in the diection o u Also illustate it using TE 6 set θ aound π This slope won t match ou eample, but the ll get 6 the idea The slope is negative hee and in eample it is suppose to be positive 39 Eample : 3 Find the diectional deivative o (, ) = 3+ 4 at the point (,0) in the diection o v =, I, o eample, (, ) epesents altitude o a mountain I ou e standing on point (,0) the diection o v 8, the ate at which (, ) is changing elevation is and move in (ascending up the mountain) 5 It s the ate at which ou unction is changing when ou move in the v diection

4 So we can ind the diectional deivative in an diection! We e standing on a mountain side and can stat walking in an diection and ind the ate o change o the teain; slope Functions o thee vaiables =,, z (gadient vecto) and diectional deivative D,, z =,, z u u u ( ) ( ) (,, ) D z is the ate o change o the unction in the diection o unit vecto u What i we want to ind the maimum change and which diection it occus? Theoem 5, page 795: Suppose is a dieentiable unction o two o thee vaiables The maimum value o the D ( ) is and it occus when u has the same diection as the diectional deivative u ( ) gadient vecto ( ) Du = u = u { cosθ Dot poduct, a b= a b cosθ and θ is between and u = cosθ = cosθ uns between and, so the MAXIMUM value o cosθ is, which occus when θ = 0 Theeoe the maimum value o Du is, and it occus when θ = 0, and since θ is between and u, it occus when u has the same diection o TE 6B is suppose to povide visual conimation o Theoem 5, but it is messed up A unction s maimum incease is o in othe wods when 0 and it occus when u has the same diection as the gadient vecto θ = I ou have a unction (, ) that epesents a mountain s suace, then is in the -plane and tells ou which diection to move on the mountain o the steepest slope The minimum change will occu when u has the eact opposite diection as the gadient vecto when θ = 80 o Fom ou eample : (, ) at (,0) = + in the diection o, The maimum ate (steepest slope) o change o (, ) (, ) = 3, 3+ So at ou point (,0), ( ) occus in the diection o to get the steepest slope and the maimum value is ( ) o, 0 = 4, 6 so that s the diection ou d move = = 5 I ou e standing on a mountain at (-, 0, 4) and ou take a step in the 4,6 o,3 diection that would be the most diicult hiking

5 Eample 3, pat a: I (, ) = e, ind the ate o change o at the point P (,0) in the diection om P to Q, Eample 3, pat b: In what diection does have the maimum ate o change? What is the maimum ate o change? See page 795 igues 7 & 8 o a visual Figue 7: The ed cuves ae level cuves to suace (, ) = e Section (topogaphical maps, isothemals, hiking map, depth o a lake ) whee level cuves o a unction o two vaiables ae the cuves with, = k, whee k is a constant equations ( ) Up on the suace at (,0) i ou move in the diection om P to Q (moving in the -plane), the slope o the tangent vecto to the suace is ; the diectional deivative is Up on the suace at (,0) i ou move in the diection, (moving in the -plane), the slope o the tangent vecto is 5 ; which is the geatest diectional deivative (slope) ou ll be able to ind on the suace at this point (,0) At (,0), the unction inceases astest in the diection o the gadient vecto (,0) =, Notice that this vecto,,, the gadient vecto, appeas to be pependicula to the level cuves though point (,0) This is not a coincidence See Figue 3 on page 798 whee all o the gadient vectos ae pependicula to the level cuves Figue 8: Shows a pictue o suace (, ) = e and gadient vecto, (moving in the -plane) in blue I am going to show ou that the gadient vecto,, is pependicula to level cuves (, ) = k See Figue on page 798

6 The quickest diection to the net level cuve is moving pependiculal to the cuve See Figue onside level cuves o a unction z (, ), ( t), ( t) I (, ) = is one o these level cuves then (, ) = z= = = = This makes sense because thee is no change in elevation i ou walk along the level cuve z = t, t Think o the map o the mountain showing elevations; i ou walked on the cuve ( () ()) ou d be staing at the eact same elevation tangent vecto () t k = But i we tun pependicula to the tangent vecto on the level cuve ou ll get the (steepest slope) z = t, t maimum ate o change o ( ( ) ( )) ( t ( ), t ( )) is the mountain teain and level cuves ( ( ), ( )) t t = kae ceated You ae eall into itness and want to get a geat wokout so ou want to hike the steepest slope up the mountain What wa should ou tun? Figue on page 798 Eplain it mathematicall: ( t ( ), t ( )) = k level cuves o (dieentiate both sides) t t d d + = 0 dt dt d d,, = 0 dt dt =, o, d d,, = 0 dt dt t = 0 () Since the dot poduct equals zeo, the gadient vecto level cuves is pependicula to the tangent vecto on the Since tangent vecto, i we shit up to the net dimension, om (,) to F(,,z), and the tangent vecto lies on the tangent plane, then F could be used as ou n omal vecto to the plane See Figue 9 on page 796! The blue suace is a level suace o F(,,z) z We consideed a 3-dimenstional unction (, ) and set it equal to a constant to see the level cuves is -dimensional ling on the -plane, pependicula to the tangent vecto on the level cuves

7 Now let s consideed a 4-dimensional unction F ( z,, ) and set it equal to a constant to see the level suaces suace F is 3-dimensional and pependicula to the tangent vecto (tangent plane) on the level Suppose a suace has equation F ( z,, ) = k, a level suace o a unction o F o thee vaiables pictued in blue in Figue 9 Let be a cuve that lies on ou suace and passes though ou point o inteest uve is descibed b continuous vecto unction t () = t ( ), t ( ), zt ( ) Since lies on S, an point must satis the equation o S so F z F ( t (), t (), zt ()) = k (level suaces o F) (dieentiate both sides) t t t F d F d F dz + + = 0 dt dt z dt F F F d d dz,,,, z dt dt dt = 0 F, F, Fz d d dz,, dt dt dt = 0 F ( t) = 0 So F is pependicula to tangent vecto and pependicula to the tangent plane and theeoe can be used as ou n omal vecto, abc,,, to the tangent plane Equation o a plane: a( 0) + b( 0) + c( z z0) = 0 Equation o a tangent plane to the level suace F ( z,, ) = kat point ( 0, 0, 0) F (,, z ): F ( ) + F ( ) + F ( z z ) = z 0 0 z has n omal vecto So in unction notation, the equation o this tangent plane is: ( )( ) ( )( ) ( )( ) F,, z + F,, z + F,, z z z = z Eample 4: z Find the equation o the tangent plane at the point (,, 3) to the ellipsoid + + = Beoe toda, we would have to solve the equation o z to ind the patial deivatives o use implicit dieentiation z z =, +, with this omula om section 4 ( )( ) ( )( ) z Now we can think o the ellipsoid as the level suace with k=3 o the unction F(,, z) = z O, i ou pee, ou can think o the ellipsoid as the level suace with k=0 o the n F(,, z) =

8 See page 797, Figue 0 o a nice visual o ou Eample 4 It would also be eas to ask o the equation o the nomal line The nomal line to a suace at a point is the line passing though the point and pependicula to the tangent plane 0 0 z z0 Hee ae the smmetic equations o an line = =, so o the a b c Nomal Line 0 0 z z0 = = F z F z F z (,, ) (,, ) (,, ) z to tangent plane We e using F = F, F, Fz as the diectional vecto to the nomal line o F See page 797, Figue 0 whee the nomal line is pictued in ed ***So, not onl gives the diection o (the steepest slope) astest incease o, it is also pependicula to the level suace S o unction though a point (Figue 9, page 796) Eample 5, pat a: Find the equation o tangent plane to z ln ( z) at point ( 0,0,) = + Beoe toda, we would have to solve the equation o z to ind the patial deivatives o use implicit dieentiation z z =, +, with this omula om section 4 ( )( ) ( )( ) Now we can think o z ln ( z) = + as the level suace when k=0 o the unction F(,, z ) = and use its gadient vecto, F, at that level suace as the nomal vecto o the tangent plane Eample 5, pat b: Find the equation o the nomal line at point ( 0,0,) to the suace z ln ( z) = +

9 A nice application o diectional vectos: Eample 6: 3 Nea a buo, the depth o a lake at the point with coodinates (, ) is z z ae measued in metes A isheman stats out at point ( 80,60 ) and moves towad the buo, located at ( 0,0 ) Is the wate unde the boat getting deepe o shallowe when he depats? = + whee,, Let s talk about making a shit to a highe level unction again: Let s ist sa we ve got a unction = ( ) (a -dimensional cuve, independent vaiable, dependent vaiable ) I ll pick = (a paabola in -dimensions) We can ewite it as 0 F, = This is a shit to a highe level unction F, is now a 3-dimensional suace with independent vaiables and and dependent vaiable ( ) F(, ) F = = and call it new unction ( ) = acts as a level cuve when 0 We can egad the paabola as a level cuve (with k=0) o F Now let s sa we ve got a unction z = (, ) k = to ou new suace F (, ) ( ) F, = = 0 (a 3-dimensional suace with independent vaiables and and dependent vaiable z) I ll pick z = + (a paaboloid in 3-dimensions) We can ewite it as z 0 F z,, = + z This is a shit to a highe level unction F z,, is now in the 4 th dimension with 3 independent vaiables,, and z and dependent vaiable ( ) F(,, z) F = + = and call it new unction ( ) 3 z = + acts as a level suace when 0 F z,, F(,, z) = + z = 0 We can egad the paaboloid as a level suace (with k=0) o F k = to ou new unction ( )