《高数双语》课件section 9.7.pptx
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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(,
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