物理

多元函数微积分

事情是这样的，有一些普物教材写了大学一年级不用知道多元函数微积分但实际上用的不亦乐乎

也有某本习题册上用jacobi行列式疯狂展开

这都没有给没学过数物方法的物竞生留一条生路

遂写一些基本的数学物理方法

笔记先传，等一会打字

$\huge{Part1.偏微商}$

设二元函数 $u = f(x, y)$，则：

$\frac{\partial u}{\partial x} = \lim_{\Delta x \to 0} \frac{f(x+\Delta x, y) - f(x, y)}{\Delta x}$

同理可得 $\frac{\partial u}{\partial y}$。

偏微分：$d_x u = \frac{\partial u}{\partial x} dx$

全微分：$du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy$

复合函数偏导

若$u = f(x, y)$，其中 $x = \varphi(t, s) , y = \psi(t, s)$

$\frac{\partial u}{\partial t} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial t}, \quad \frac{\partial u}{\partial s} = \frac{\partial u}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial u}{\partial y}\frac{\partial y}{\partial s}$

若$u = f(x_1, \dots, x_n)$ ，$x_i = f_i(t_1, \dots, t_m)$

$\frac{\partial u}{\partial t_k} = \sum_{i=1}^n \frac{\partial u}{\partial x_i} \frac{\partial x_i}{\partial t_k}$

若$u = f(x, y, z)$ ，$y = \varphi(x, t), z = \psi(x, t)$

$ \frac{\partial u}{\partial x} = f'_x + f'_y \cdot \varphi'_x + f'_z \cdot \psi'_x$

$\frac{\partial u}{\partial t} = f'_y \cdot \varphi'_t + f'_z \cdot \psi'_t$

若$u = f(x_1, \dots, x_n)$ ，$x_i = x_i(t)$

$\frac{du}{dt} = \sum_{i=1}^n \frac{\partial u}{\partial x_i} \frac{dx_i}{dt}$

Euler公式（齐次函数)

若 $f(tx, ty, tz) = t^k f(x, y, z)$，则：

$x\frac{\partial f}{\partial x} + y\frac{\partial f}{\partial y} + z\frac{\partial f}{\partial z} = kf$