向量函數的微分 (Derivative of vector-values function)

  • 令函數f:(a,b)Rn\mathbf{f}: (a,b) \rightarrow \mathbb{R}^{n}, f=(f1,f2,,fn)\mathbf{f} = (f_1, f_2, \cdots, f_n) and fk:(a,b)R, kf_k: (a,b) \rightarrow \mathbb{R}, \ \forall k.
    • f\mathbf{f}在點c(a,b)c \in (a,b)可微若fk(c)<, kf_k^{'}(c) < \infty,\ \forall k,即每一個維度都必須可微分.
    • f(c)=(f1(c),f2(c),,fn(c)\mathbf{f}^{'}(c) = (f_1^{'}(c), f_2^{'}(c), \cdots, f_n^{'}(c).

偏微分(Partial derivative)

  • SRnS \subseteq \mathbb{R}^{n}為開集合,函數f:SRf: S \rightarrow \mathbb{R},令x,cS\mathbf{x, c} \in S, x=(x1,x2,,xn), c=(c1,c2,,cn)\mathbf{x} = (x_1, x_2, \cdots, x_n), \ \mathbf{c} = (c_1, c_2, \cdots, c_n),且兩向量間只有第kk個維度不同,其它維度均相同(即xi=ci, ikx_i = c_i, \ \forall i \neq kxkckx_k \neq c_k),則對第kk個維度的偏微分如下:
    • fxk(c)=limxkckf(x)f(c)xkck \frac{\partial f}{\partial x_k}(\mathbf{c}) = \lim_{x_k \rightarrow c_k} \frac{f(\mathbf{x}) - f(\mathbf{c} )}{x_k - c_k} .
    • 其它常用符號為Dkf(c) D_k f(\mathbf{c}).

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