若∫在开集D的每一点都连续可微，则称∫在D上一阶连 续可微；若∫在开集D的每一点都都二阶连续可微，则称∫在D 上二阶连续可微 由上述定义不难发现，若∫在x二阶连续可微，则 82f(x)82f(x) 0x;∂xj i,j=1,2,·,n 8ioxi 即Hesse阵V2f(x)是对称阵 例1.1设二次函数 K)-detinn 其中c∈R”,H∈Rnxm是对称阵.那么，不难计算它在x的梯度及 Hesse阵为 Vf(x)=c+Hx,V2f(x)=H. Back Close
11/36 JJ II J I Back Close e f 3m8 D zò:—ÎYåáßK° f 3 D ˛ò Î Yåá¶e f 3m8 D zò:—— ÎYåáßK° f 3 D ˛ ÎYåá. d˛„½¬ÿJuyße f 3 x ÎYåáßK ∂ 2 f(x) ∂xi∂xj = ∂ 2 f(x) ∂xj∂xi , i, j = 1, 2, · · · , n, = Hesse ∇2 f(x) ¥È° . ~ 1.1 gºÍ f(x) = c T x + 1 2 x THx, Ÿ• c ∈ R n , H ∈ R n×n ¥È° . @oßÿJOéß3 x F›9 Hesse è ∇f(x) = c + Hx, ∇2 f(x) = H
例1.2（泰勒展开）.设函数f:Rn→R连续可微，那么 f(x+h)f(a)+Vf(x+rh)Thdr f(x)+Vf(a+Eh)Th,EE(0,1) =f(x)+Vf(x)"h+o(lhl). 进一步，若函数∫是二次连续可微的，则有 r+)=f(r)+Vf(h+f(+hudr =f四+Vfh+v2fr+hh,∈0，) F(a)+VF(a)"h+hV"f(r)h+olh) Back Close
12/36 JJ II J I Back Close ~ 1.2 (V–m). ºÍ f : R n → R ÎYåáß@o f(x + h) = f(x) + Z 1 0 ∇f(x + τh) T hdτ = f(x) + ∇f(x + ξh) T h, ξ ∈ (0, 1) = f(x) + ∇f(x) T h + o(khk). ?ò⁄, eºÍ f ¥gÎYåá, Kk f(x + h) = f(x) + ∇f(x) T h + Z 1 0 (1 − τ )h T∇2 f(x + τh)hdτ = f(x) + ∇f(x) T h + 1 2 h T∇2 f(x + ξh)h, ξ ∈ (0, 1) = f(x) + ∇f(x) T h + 1 2 h T∇2 f(x)h + o(khk 2 )