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：

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

隐函数

由 $F(x_1, \dots, x_n) = 0 $ 确定 $x_n = x_n(x_1, \dots, x_{n-1})$，则：

$\frac{\partial x_n}{\partial x_i} = - \frac{\frac{\partial F}{\partial x_i}}{\frac{\partial F}{\partial x_n}} \quad (\frac{\partial F}{\partial x_n} \neq 0)$

n阶全微分

$d^n u = \left( dx_1 \frac{\partial}{\partial x_1} + dx_2 \frac{\partial}{\partial x_2} + \dots + dx_n \frac{\partial}{\partial x_n} \right)^n u$

$\huge{Part2.二重积分}$

雅克比行列式

对于变换 $y_i = f_i(x_1, \dots, x_n)$，雅可比行列式记为：

$J = \frac{\partial(y_1, y_2, \dots, y_n)}{\partial(x_1, x_2, \dots, x_n)} = \begin{vmatrix} \frac{\partial y_1}{\partial x_1} & \cdots & \frac{\partial y_1}{\partial x_n} \\ \vdots & \ddots & \vdots \\ \frac{\partial y_n}{\partial x_1} & \cdots & \frac{\partial y_n}{\partial x_n} \end{vmatrix}$

若 $y_i$是 $x_j$ 的函数，$x_j$ 是 $t_k$ 的函数，则：

$\frac{\partial(y_1, \dots, y_m)}{\partial(t_1, \dots, t_m)} = \sum_{(i_1, \dots, i_m)} \frac{\partial(y_1, \dots, y_m)}{\partial(x_{i_1}, \dots, x_{i_m})} \frac{\partial(x_{i_1}, \dots, x_{i_m})}{\partial(t_1, \dots, t_m)}$

二重积分

定义：$\iint_D f(x, y) dxdy = \lim_{\lambda \to 0} \sum_{i,j} f(\xi_i, \eta_j) \Delta x_i \Delta y_j$

直角坐标系计算$\int_a^b dx \int_{\varphi_1(x)}^{\varphi_2(x)} f(x, y) dy$

极坐标变换：$x = \rho\cos\varphi, y = \rho\sin\varphi$

雅可比行列式 $J = \rho$，故 $dx dy = \rho d\rho d\varphi$。

$\iint_D f(x, y) dx dy = \iint_{D'} f(\rho\cos\varphi, \rho\sin\varphi) \rho d\rho d\varphi$

变量替换：$ x=x(u,v), y=y(u,v) $

$\iint_D f(x, y) dx dy = \iint_{D'} f[x(u,v), y(u,v)] |J| du dv$