问 . 何谓雅可比矩阵? 它有什么意义?
答：首先讨论 \mathrm{n} 元函数 y=f(x) （ x \in A \subset R^n, y \in R ) 在点 x=\left(x_1, x_2, \cdots, x_n\right) 的全微分。
若函数 y=f(x) 在点 \mathrm{x} 可微, 则函数 y=f(x) 在点 \mathrm{x} 的全微分
d y=\frac{d f}{\partial x_1} d x_1+\frac{d f}{\partial x_2} d x_2+\cdots+\frac{d f}{\partial x_n} d x_n,
\mathrm{dy} 可写成向量形式
d y=\left(\frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \cdots, \frac{\partial f}{\partial x_n},\left(\begin{array}{c}
d x_1 \\
d x_2 \\
\vdots \\
d x_n
\end{array}\right)\right.
其次讨论一元向量函数 y=f(x)\left(x \in(a, b) \subset R, y \in R^n\right) 在点 \mathrm{x} 的全微分, 其中 y=\left(y_1, y_2, \cdots y_n\right), f=\left(f_1, f_2, \cdots, f_n\right) 。
若每个函数
\frac{\partial\left(f_1, \cdots, f_n\right)}{\partial\left(x_1, \cdots, x_n\right)}=\left|\begin{array}{cccc}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\
\vdots & \vdots & & \\
\frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \cdots & \frac{\partial f_n}{\partial x_n}
\end{array}\right| f_k \text { 在点 } \mathrm{x} \text { 可微 }(k=1,2, \cdots, n) , 则函数 y=f(x) 在点 \mathrm{x} 的全微分
d y=\left(\begin{array}{c}
d y_1 \\
d y_2 \\
\vdots \\
d y_n
\end{array}\right)=\left(\begin{array}{c}
f_1^{\prime}(x) d x \\
f_2^{\prime}(x) d x \\
\vdots \\
f_n^{\prime}(x) d x
\end{array}\right)=\left(\begin{array}{c}
f_1^{\prime}(x) \\
f_2^{\prime}(x) \\
\vdots \\
f_n^{\prime}(x)
\end{array}\right) d x
最后讨论向量函数 y=f(x)\left(x \in A \subset R^n, y \in R^m\right) 在点 \mathrm{x} 的全微分, 其中 x=\left(x_1, x_2, \cdots, x_n\right), y=\left(y_1, y_2, \cdots, y_m\right), f=\left(f_1, f_2, \cdots, f_m\right) 。
若每个函数 f_k 在点 \mathrm{x} 可微 (k=1,2, \cdots, n), 则向量函数 y=f(x) 在点 \mathrm{x} 全微分
d y=\left(\begin{array}{c}
d y_1 \\
d y_2 \\
\vdots \\
d y_m
\end{array}\right)=\left(\begin{array}{c}
\frac{\partial f_1}{\partial x_1} d x_1+\frac{\partial f_1}{\partial x_2} d x_2+\cdots+\frac{\partial f_1}{\partial x_n} d x_n \\
\frac{\partial f_2}{\partial x_1} d x_1+\frac{\partial f_2}{\partial x_2} d x_2+\cdots+\frac{\partial f_2}{\partial x_n} d x_n \\
\vdots \vdots \\
\frac{\partial f_m}{\partial x_1} d x_1+\frac{\partial f_m}{\partial x_2} d x_2+\cdots+\frac{\partial f_m}{\partial x_n} d x_n
\end{array}\right)
将 dy 改写成矩阵形式是
d y=\left(\begin{array}{ccc}
\frac{\partial f_1}{\partial x_1} \frac{\partial f_1}{\partial x_2} & \cdots \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} \frac{\partial f_2}{\partial x_2} & \cdots \frac{\partial f_2}{\partial x_n} \\
\vdots & \vdots & \vdots \\
\frac{\partial f_m}{\partial x_1} \frac{\partial f_m}{\partial x_2} & \cdots \frac{\partial f_m}{\partial x_n}
\end{array}\right)\left(\begin{array}{c}
d x_1 \\
d x_2 \\
\vdots \\
d x_n
\end{array}\right)
定义 若向量函数 y=f(x)\left(x \in A \subset R^n, y \in R^m\right), 即
\left\{\begin{array}{c}
y_1=f_1\left(x_1, x_2, \cdots, x_n\right) \\
y_2=f_2\left(x_1, x_2, \cdots, x_n\right) \\
\cdots \cdots \\
y_n=f_n\left(x_1, x_2, \cdots, x_n\right)
\end{array}\right.
有连续的偏导数, 则
\left(\begin{array}{c}
\frac{\partial f_1}{\partial x_1} \frac{\partial f_1}{\partial x_2} \cdots \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} \frac{\partial f_2}{\partial x_2} \cdots \frac{\partial f_2}{\partial x_n} \\
\vdots \\
\frac{\partial f_m}{\partial x_1} \frac{\partial f_m}{\partial x_2} \cdots \frac{\partial f_m}{\partial x_n}
\end{array}\right)
称为 m \times n_{\mathrm{m}} 雅可比矩阵, 表为 f^{\prime}(x) 。
如果 d x=\left(d x_1, d x_2, \cdots d x_n\right), 向量函数 y=f(x)\left(x \in A \subset R^n, y \in R^m\right) 在点 \mathrm{x} 有连续偏导数, 则向量函数 y=f(x) 在点 \mathrm{x} 可微, 且 d y=f^{\prime}(x) d x \text {. }
从形式上说, 向量函数的微分与一元函数 (R \rightarrow R) 的微分完全相同, 雅可比矩阵 $f^{\prime}(x)$就相当于一元函数的导数 f^{\prime}(x) 。
特别是, 当 \mathrm{n}=\mathrm{m} 时, 雅可比矩阵就变成了雅可比方阵, 即函数行列式:
\frac{\partial\left(f_1, \cdots, f_n\right)}{\partial\left(x_1, \cdots, x_n\right)}=\left|\begin{array}{cccc}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_n} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_n} \\
\vdots & \vdots & & \\
\frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \cdots & \frac{\partial f_n}{\partial x_n}
\end{array}\right|
在向量函数微分学的理论中, 雅可比矩阵的作用类似于一元函数微分学的理论中导数所起的作用。
欢迎大家进行讨论~