微积分全英微积分全英 (40).pdf

12.5 Directional Derivatives and GradientsProblem IntroductionQ1:How to define directional derivative?Q2:Whats the connection with gradient?DefinitionIf the limit exists,y P x xyOPthen the limit value is called the directional derivative of +0()()(,)(,)limlimPPf Pf Pf xx yyf x y at the point in the direction ,|,PP where 22xy()().Definition y P x xyOP+0(,)(,)limff xx yyf x yl Denote it flRemark.The directional derivative is the rate of change of the function along the direction of half line.Theorem where angle between the direction and the positive laxis.x angle between the direction and the positive laxis.ycoscosffflxy，Let be differentiable at .Then has a directional derivative at in the any given direction ,(,)zf x y(,)P x y(,)P x y(,)zf x yProof of the theorem ),(),(yxfyyxxf)(oyyfxxf cos cos ),(),(yxfyyxxf )(oyyfxxf Proof:Since is differentiabler,the increment can be representedDivide both sides by at the same time,we get differentiable=cos+cos y P x xyOPProof of the theorem y P x xyOPSo,the directional derivative ),(),(lim0yxfyyxxf flcoscos.ffxyNote(1)coscosffflxywhere are the directional cosines of the direction,cos,cos and are the directional angles.,0,partial derivatives of the function and the direction .l(2)Calculating the directional derivative only needs to know the Note(3)Relations The function is differential The function has the directional derivative.Counterexample:(,)=2+2,is an arbitrary direction passing through(0,0),directional derivative exist:(0,0)=lim0+(0+cos,0+cos)(0,0)=lim0+|0=1,But the partial derivative does not exist:(0,0)=lim0(,0)(0,0)=lim0|,So,it is not differentiable.Example 1If =2+2,find the directional derivative of at 0(1,2)in the direction ,where is the the vector from 0 to (2,2+3).=0=(1,3),cos=12,cos=32,Two partials are (1,2)=2,(1,2)=4,so the directional derivative 0=(1,2)cos+(1,2)cos=2 12+4 32=1+2 3.Example 2 If ,find the directional derivative atin the direction ,where is the angle between and Moreover,find these directions such that the directional derivative has(1)maximum value;(2)zero.22(,)fx yxxyy(1,1)Plaxis.xl,sin)2(cos)2()1,1()1,1(xyyx sincos 2 sin()4 sin)1,1(cos)1,1()1,1(yxfflf By the calculation formula of the directional derivative,Example 2(1,1)Plaxis.xl(1)When =4(2)When 3=4 If ,find the directional derivative atin the direction ,where is the angle between and Moreover,find these directions such that the directional derivative has(1)maximum value;(2)zero If ,find the directional derivative atin the direction ,where is the angle between and Moreover,find these directions such that the directional derivative has(1)maximum value;(2)zero.22(,)fx yxxyy(1,1)Plaxis.xl the directional derivative arrives the maximum 2.the directional derivative is equal to zero.2 sin()(1,1)4fl Generalize the calculation formula for the function of three variables.Extension Similarly,if the function is differential at some point,then it has the directional derivative at the point in any direction.(cos,cos,cos)is the directional vector of .lVector form Vector form:=(,),unitvector =(cos,cos),passing through point P,()lim0(+)();If f is differentiable at P,then()=().Connection with the GradientThe calculation formula In what direction does have the maximal directional derivative?(,)zf x y are in the same directions,the maximum valueWhen()and =()=|()|cos(),)=cos+cosmax =|()|.Connection with the GradientConnection with the Gradient=()=|()|cos(),)=cos+cosGradient:()=(),()=()+()(=(,)()is also denoted by grad f(p)or grad|Conclusion(1)is the maximum value of the directional derivative.The module of the gradient The gradient of a function at a certain point is a vector:direction the direction of the maximum directional derivative module the maximum value of the directional derivative.Connection with the GradientConnection with the GradientConclusion(2)The directional derivative is regarded as a functional of direction:When and()are in the same directions,has the maximum|()|;i.e.inthedirection of()，the function is changing most rapidly;When and()are in the opposite directions,has the minimum|()|;When (),=0.Example 3If (1)find the rate of change in the direction from to .(2)find the direction with maximal growth rate at and try to find the maximal growth rate.(,)yzfx yxe(2,0)P1(,2)2Q(2,0)P,53cos ,54cos (2,0)|fl(1)=(,),()=(,)|=(,),()(cos,cos)=(1,2)35,45=1.Example 3If (1)find the rate of change in the direction from to .(2)find the direction with maximal growth rate at and try to find the maximal growth rate.(,)yzfx yxe(2,0)P1(,2)2Q(2,0)P(2)The maximal growth rate is max =|(2,0)|=|1,2|=5.The corresponding direction is 1,2.Note Find second the four partial derivatives of If ,find the directional derivative at in the direction from to .2 yzxe(1,0)PThe differences of the definition between the directional derivative and partial derivative must be positive!1.The directional derivative is the rate of change of the function at certain point in any direction.lNote Find second the four partial derivatives of If ,find the directional derivative at in the direction from to .2 yzxe(1,0)PThe differences of the definition between the directional derivative and partial derivativexyxfyxxfxfx ),(),(lim0yyxfyyxfyfy ),(),(lim0 They are the rate of change of straight lines parallel to the axises ofa function at a certain point.x、y can be positive or negative!2.However,the partial derivatives SummaryThe concept of directional derivativeThe concept of the gradientThe relation between the directional derivative and gradient Questions and AnswersIf ,find the gradient and try to find these point whose gradient are zero.2222332u xyzxy1,1,2,6)24()32(kzjyix kzujyuixuzyxgradu ),(gradSo,.1225)2,1,1(kjigradu gradThe gradient is zero at 03 1(,0)2 2P By the calculation formula of the gradient,Directional Derivatives and Gradients DefinitionCalculation formula That is,coscos(,)cos,cosfffgradf x ylxy Let have the partial derivative at the point ,thenthe vector is called the gradient of at the point .),(yxfz (,)P x y),(yxfz (,)P x yGradient(,)fffgradf x y zijkxyzmodule the maximum value of the directional derivativedirection direction of the maximum directional derivative has first-order continuous partial derivatives in the space area For any point),(zyxfu ,G,),(GzyxP The gradient is also a vector: