The Gradient and Applications This unit is based on Sections 9.5 and 9.6, Chapter 9. All assigned readings and exercises are from the textbook

Transcription

1 The Gadient and Applicatins This unit is based n Sectins 9.5 and 9.6 Chapte 9. All assigned eadings and eecises ae fm the tetbk Objectives: Make cetain that u can define and use in cntet the tems cncepts and fmulas listed belw: 1. find the gadient vect at a given pint f a functin.. undestand the phsical intepetatin f the gadient. 3. find a multi-vaiable functin given its gadient 4. find a unit vect in the diectin in which the ate f change is geatest and least given a functin and a pint n the functin. 5. the ate f change f a functin in the diectin f a vect. 6. find nmal vect and tangent vects t a cuve 7. wite equatins f the tangent line and the nmal line. 8. find an equatin f the tangent plane t a suface Reading: Read Sectin 9.5 pages Eecises: Cmplete pblems 9.5 and 9.6 Peequisites: Befe stating this Sectin u shuld... familia with the cncept f patial diffeentiatin be familia with vect functins 1

2 Diectinal Deivatives: definitin Cnside the tempeatue T at vaius pints f a heated metal plate. Sme cntus f T ae shwn in the diagam. T15 T0 B A T5 We ae inteested in hw T changes fm ne pint t anthe.

3 Diectinal Deivatives: definitin The ate f change f T in the diectin specified b AB is given b: 0-15/AB 5/h an eamples f diectinal deivative A B T15 T5 T0 In geneal f a given functin T T the diectinal deivative in the diectin f a unit vect u < csθ sinθ> is D u T lim h 0 T hcsθ hsinθ h T whee h 3

4 Diectinal Deivatives: definitin A the gadient vect at a pint A magnitude the lagest diectinal deivative and pinting in the diectin in which this lagest diectinal deivative ccus is knwn as the gadient vect. 4

5 The gadient vect: gad A vect field called the gadient witten: gad can be assciated with a scala field. At eve pint the diectin f the vect field is thgnal t the scala field cntu C and in the diectin f the maimum ate f change f. ˆi ˆ j kˆ called del 5

6 6 The gadient vect: gad gad w k j i Given : Gadient f a unctin ind : Given / 3 Eample:

7 Ke pints: The gadient vect Cnt. diectin is the nmal vect t the suface tˆ 0 tˆ is a tangent unit vect t magnitude gives the ate f change the slpe ate f change f when mving alng a cetain diectin. pints in the diectin f mst apid incease f. - pints in the diectin f mst apid decease f. is a scala field while is a vect field. is nt cnstant in space. T 7

8 The gadient vect Cnt. Genealiatin f Diectinal Deivative in u diectin: C tˆ u D u uˆ uˆ unit vect The maimum value f D u is and ccus when u and ae in the same diectins. The minimum value f D u is - and ccus when u and ae in the ppsite diectins. Eample: / ; u < 68 > ind 1. u 8

9 The gadient vect: Sme Applicatins Enginees use the gadient vect in man phsical laws such as: 1. Electic ield E and Electic Ptential V: E V. Heat lw H and Tempeatue T: H k T k cnstant 3. ce ield and Ptential Eneg U: U Eample Given : T 100 ind the heat flw vect H 9

10 Tempeatue Distibutin: Clest What T lcatin : cente diectin the insect Insect Eample! 5 Assuming the insect is familia with vect analsis and knws the tempeatue distibutin!! shuld g t cl ff the fastet? d T 4i j -16i-4 j 10

11 11 Equatin f Tangent Plane A vect nmal t the suface c at a pint P is and can be dented b n. If is the psitin vect f the pint P elative t the igin and is the psitin vect f an pint n the tangent plane the vect equatin f the tangent plane is: P at n n 0 0 The equivalent scala equatin f the tangent plane is: Eample: 1 9 : : Pint Suface

12 1 Equatin f Nmal Line t a Suface A vect nmal t the suface c at a pint P is n. If is the psitin vect f the pint P and is the psitin vect f an pint n the nmal line the vect equatin f the nmal line t the suface is: P at n n 0 t t t The equivalent paametic equatin f the nmal line is: Eample: : : Pint Suface