1、Section 9.712RDirectional Derivatives and the Gradient02xR,2R,l is a vector in the plane 02:(x)RR.fUWe draw the straight line L through the point x0 in the parallel to l,whose equation is0 xxe,R.ltt0 xxeltLine0 xlelDirection of increasing t,xyOThe rate of change of the function f 0 xis just the rate
2、 of change of f atin the direction l at the point 0 xmoves with motion restricted0 xwhen to the line L.,lIe 3Directional Derivatives and the Gradientis actually a function of a single variable t,0(x)(xe)lfftfunction denoted by 0()(xe).lF tftWhen varies on the line L,the xeland are both fixed and the
3、 point 0 xDefinition (Directional Derivative 方向导数方向导数)is the0 xThe derivative of f at el in the direction of the unit vector number 00000 x(xe)(x)()(0)limlim.llttftffF tFtt in the direction l.0 xthis value is called the directional derivative of f at or0(x).lf denoted by 0 xlf 4Directional Derivativ
4、es and the GradientFind the directional 2222422,0,(,)0,0.xyxyxyf x yxy Example Letderivative of the function f at the point(0,0)in the direction e(cos,sin).l Solution we have cos0,If (0,0)lf 22240cos sinlimcossintt 0(cos,sin)(0,0)limtf ttft 2sin;cos 5Directional Derivatives and the GradientSolution(
5、continued)we have cos0,If 2222422,0,(,)0,0.xyxyxyf x yxy (0,0)lf 0(cos,sin)(0,0)limtf ttft 0.Note It is easy to see that in last(0,0)2l2f ;4 as(0,0)2l2f .4 asIn general,it is easy to see00 xx.(l)lff 6Directional Derivatives and the GradientTheorem(Formula for the directional derivative)Suppose thatT
6、hen the function(,)zf x y is differentiable at the point 00(,).xy00(,)xy in any direction l exists,the directional derivative at the point and000000(,)(,)(,)coscos,lxyxyxyfffxyis an unit vector in the direction l so that e(cos,cos)l where,are the direction angles of l.7Directional Derivatives and th
7、e Gradient000000(,)(,)(,)coscos,lxyf xyf xyfxy Proof By the definition of directional derivative,we have 0000(,)(,)f xx yyf xy 220000(,)(,)()().xyfxyxfxyyoxy Then the increments of thee(cos,cos).l We choose a direction We notice that may be written as,y independent variables x and y,x andcosxt cos,y
8、t andrespectively.2222()()(cos)(cos)|.xyttt Then,we have8Directional Derivatives and the Gradient0000(,)(,)f xx yyf xy220000(,)(,)()()xyfxyxfxyyoxy Proof(continued)0000(cos,cos)(,)f xtytf xy0000(,)(,)().xyfxyxfxyyo t so that00000(cos,cos)(,)limltf xtytf xyft 00000()lim(,)cos(,)cosxyto tfxyfxyt0000(,
9、)cos(,)cos.xyfxyfxyNote This conclusion can be easily to extended to functions of n For example,variables.0000 x(x)cos(x)cos(x)cos,lxyzffff where e(cos,cos,cos)l is a unit vector in the direction l.9Directional Derivatives and the GradientFind the directional derivative 2(,).yzf x yxeExample Suppose
10、 thatof function f at the point P(1,0)in the direction from point P(1,0)topoint Q(2,-1).Solution then the included l1,1,PQ The direction vector here is angle between l and the x-axis is11,(,).422le 2(1,0)(1,0)1yzex 2(1,0)(1,0)22,yzxey andthen the directional derivative is11222(1,0)zl 2.2 000000(,)(,
11、)cos(,)coslxyxyffxyfxy 10Directional Derivatives and the GradientThe formula of directional derivativecan also be written as0000 xxxxcoscoscoslffffxyz is the inner product.0000g(x)(x),(x),(x)xyzfff where,is a vector and 00 xg(x),e,llf 00 xg(x),e|g|cos(g,e),lllf By the definition of inner product,we
12、notice thatThis means that the directional derivative of function f at point x0 in the direction of l is just the projection vector of g onto the unit vector el.11Directional Derivatives and the GradientDefinition 1(Gradient)Suppose that the function of three variables 000 x(,).xy Then the(,)uf x y
13、is differentiable at the point 00 xx,ffxyis called the gradient vector of f at x0,vectororor gradient for short.0(x),f This vector is denoted by 0grad(x)fnamely 0000 xxgrad(x)(x),.ffffxy Here,xy read as del12Directional Derivatives and the GradientExample Finding Directions of Maximal,Minimal,and Ze
14、ro Change(a)Increases most rapidly at the point(1,1).SolutionFind the direction in which 22(,)(/2)(/2)f x yxy(c)What are the directions of zero change in f at(1,1).(b)Decreases most rapidly at(1,1).(a)Increases most rapidly at the point(1,1).at(1,1).f The function increases most rapidly in the direc
15、tion of The gradient there is (1,1)f 1,1.(1,1),ffxy (1,1),x y 13Directional Derivatives and the GradientSolution(b)Decreases most rapidly at(1,1).The function decreases most rapidly in the direction of fat(1,1),which isf(1,1).The directions of zeros change at(1,1)are the directions orthogonalto:f(-1
16、,1)and (1,-1).(c)What are the directions of zero change in f at(1,1).14Directional Derivatives and the GradientDefinition 2(Gradient)Suppose that the function of three variables 0000123x(,).xxx Then the123(,)uf xxx is differentiable at the point 000123xxx,fffxxxis called the gradient vector of f at
17、x0,vectororor gradient for short.0(x),f This vector is denoted by 0grad(x)fnamely 00000123xxxgrad(x)(x),.fffffxxx Here 123,xxx read as del15Directional Derivatives and the GradientRules for operations on gradientsBy the rules of derivation,it is easy to obtain some rules for operationson gradients as follows:or()().f ufuu(4)grad()()gradf ufuu or1212();C uC vCuCv (1)1212grad()gradgradC uC vCuCvor();uvu vv u (2)grad()gradgraduvuvvuor2,0;uv uu vvvv (3)21grad(gradgrad)uvuuvvv
