新讲 第12章 多元微分学 第6题
📝 题目
例 6 对于更一般的 $m$ 元函数,考虑与上例类似的问题: 设 $f\left( x\right) = f\left( {{x}_{1},{x}_{2},\cdots ,{x}_{m}}\right)$ 是 $n$ 阶连续可微的函数,记
$$ \varphi \left( t\right) = f\left( {x + {th}}\right) $$
$$ = f\left( {{x}_{1} + t{h}_{1},{x}_{2} + t{h}_{2},\cdots ,{x}_{m} + t{h}_{m}}\right) . $$
试计算 ${\varphi }^{\left( n\right) }\left( t\right)$ .
💡 答案解析
解 设 $g\left( x\right) = g\left( {{x}_{1},{x}_{2},\cdots ,{x}_{m}}\right)$ 是连续可微函数,记
$$ \psi \left( t\right) = g\left( {x + {th}}\right) $$
$$ = g\left( {{x}_{1} + t{h}_{1},{x}_{2} + t{h}_{2},\cdots ,{x}_{m} + t{h}_{m}}\right) . $$
与上例中类似,我们注意到: 以 $\frac{\mathrm{d}}{\mathrm{d}t}$ 作用于 $\psi \left( t\right)$ ,相当于以算子
$$ \left( {{h}_{1}\frac{\partial }{\partial {x}_{1}} + {h}_{2}\frac{\partial }{\partial {x}_{2}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}}\right) $$
作用于
$$ g\left( {{x}_{1} + t{h}_{1},{x}_{2} + t{h}_{2},\cdots ,{x}_{m} + t{h}_{m}}\right) . $$
运用这一观察结果, 我们求得
$$ {\varphi }^{\prime }\left( t\right) = \left( {{h}_{1}\frac{\partial }{\partial {x}_{1}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}}\right) f\left( {x + {th}}\right) , $$
$$ {\varphi }^{\prime \prime }\left( t\right) = {\left( {h}_{1}\frac{\partial }{\partial {x}_{1}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}\right) }^{2}f\left( {x + {th}}\right) , $$
..................................................
$$ {\varphi }^{\left( n\right) }\left( t\right) = {\left( {h}_{1}\frac{\partial }{\partial {x}_{1}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}\right) }^{n}f\left( {x + {th}}\right) . $$
对于连续可微足够多次的函数,求偏导数的运算 $\frac{\partial }{\partial {x}_{1}},\cdots ,\frac{\partial }{\partial {x}_{m}}$ 可以互相交换顺序. 涉及 $\frac{\partial }{\partial {x}_{1}},\cdots ,\frac{\partial }{\partial {x}_{m}}$ 这些算子的相加,相乘以及乘以实数的运算, 遵循多项式代数中有关文字符号的运算法则. 因此, 对
$$ {\left( {h}_{1}\frac{\partial }{\partial {x}_{1}} + {h}_{2}\frac{\partial }{\partial {x}_{2}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}\right) }^{n} $$
可以按照代数中的“多项式定理”(参看本节后的补充内容)予以展开:
$$ {\left( {h}_{1}\frac{\partial }{\partial {x}_{1}} + {h}_{2}\frac{\partial }{\partial {x}_{2}} + \cdots + {h}_{m}\frac{\partial }{\partial {x}_{m}}\right) }^{n} $$
$$ = \mathop{\sum }\limits_{{{p}_{1} + \cdots + {p}_{m} = n}}\frac{n!}{{p}_{1}!\cdots {p}_{m}!}{h}_{1}^{{p}_{1}}\cdots {h}_{m}^{{p}_{m}}\frac{{\partial }^{n}}{\partial {x}_{1}^{{p}_{1}}\cdots \partial {x}_{m}^{{p}_{m}}}, $$
这里
$$ \frac{n!}{{p}_{1}!{p}_{2}!\cdots {p}_{m}!}\;\left( {{p}_{1} + {p}_{2} + \cdots + {p}_{m} = n}\right) $$
是多项式系数.