高等数学 A（下）公式

BUPT 高等数学 A（下）课程涉及的公式，自制

填空题

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\[多元偏导数连续 [...] 多元函数可微\]	\[多元偏导数连续、rightarrow 多元函数可微\]
\[多元函数可微 [...] 多元函数沿任意方向导数均存在\]	\[多元函数可微、rightarrow 多元函数沿任意方向导数均存在\]
\[多元函数可微 [...] 多元函数偏导数存在\]	\[多元函数可微、rightarrow 多元函数偏导数存在\]
\[多元函数可微 [...] 多元函数连续\]	\[多元函数可微、rightarrow 多元函数连续\]
\[多元函数偏导数存在 [...] 多元函数连续\]	\[多元函数偏导数存在、nleftrightarrow 多元函数连续\]
\[多元函数偏导数存在 [...] 多元函数沿任意方向导数存在\]	\[多元函数偏导数存在、nleftrightarrow 多元函数沿任意方向导数存在\]
\[多元函数连续 [...] 多元函数沿任意方向导数存在\]	\[多元函数连续、nleftrightarrow 多元函数沿任意方向导数存在\]
函数\(z=f(x,y)\)在\((x_0,y_0)\)具有偏导数的必要条件：令\(A=f_{xx}(x_0,y_0),B=f_{xy}(x_0,y_0),C=f_{yy}(x_0,y_0)\)，则\(AC-B^2>0\)时 [...]，\(AC-B^2<0\)时 [...]，\(AC-B^2=0\)时 [...]	函数\(z=f(x,y)\)在\((x_0,y_0)\)具有偏导数的必要条件：令\(A=f_{xx}(x_0,y_0),B=f_{xy}(x_0,y_0),C=f_{yy}(x_0,y_0)\)，则\(AC-B^2>0\)时具有极值，且当\(A<0\)时有极大值，\(A>0\)时有极小值，\(AC-B^2<0\)时没有极值，\(AC-B^2=0\)时可能有极值，也可能没有
\[\sum_{n=0}^{\infty}aq^n\left\{\begin{align*}[...] & |q| < 1 \\ [...] & |q|\geq 1\end{align*}\right.\]	\[\sum_{n=0}^{\infty}aq^n\left\{\begin{align*} \text{收敛} & |q| < 1 \\ \text{发散} & |q|\geq 1\end{align*}\right.\]
\[\sum_{n=0}^{\infty}\frac{1}{n^p}\left\{\begin{align*}[...] & p > 1 \\ [...] & p\leq 1\end{align*}\right.\]	\[\sum_{n=0}^{\infty}\frac{1}{n^p}\left\{\begin{align*} \text{收敛} & p > 1 \\ \text{发散} & p\leq 1\end{align*}\right.\]
设\(f(x)\)是周期为\(2\pi\)的周期函数，傅里叶级数\[[...]\]\[a_0=[...]\]\[a_n=[...]\]\[b_n=[...]\]	设\(f(x)\)是周期为\(2\pi\)的周期函数，傅里叶级数\[ f(x)=\frac{a_0}2+\sum_{n=1}^{\infty}(a_n\cos nx+b_n\sin nx) \]\[a_0= \frac1\pi\int_{-\pi}^{\pi}f(x)\mathop{}\!\mathrm{d}x \]\[a_n= \frac1\pi\int_{-\pi}^{\pi}f(x)\cos nx\mathop{}\!\mathrm{d}x \]\[b_n= \frac1\pi\int_{-\pi}^{\pi}f(x)\sin nx\mathop{}\!\mathrm{d}x \]
设\(f(x)\)是周期为\(2l\)的周期函数，傅里叶级数\[[...]\]\[a_0=[...]\]\[a_n=[...]\]\[b_n=[...]\]	设\(f(x)\)是周期为\(2l\)的周期函数，傅里叶级数\[ f(x)=\frac{a_0}2+\sum_{n=1}^{\infty}(a_n\cos \frac{n\pi}l x+b_n\sin \frac{n\pi}l x) \]\[a_0= \frac1l\int_{-l}^{l}f(x)\mathop{}\!\mathrm{d}x \]\[a_n= \frac1l\int_{-l}^{l}f(x)\cos \frac{n\pi}l x\mathop{}\!\mathrm{d}x \]\[b_n= \frac1l\int_{-l}^{l}f(x)\sin \frac{n\pi}l x\mathop{}\!\mathrm{d}x \]

基础

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设函数\(F(x,y)\)在点\(P(x_0,y_0)\)的某一邻域内具有连续偏导数，且\(F(x_0,y_0)=0,F_y(x_0,y_0)\not= 0\)，则方程\(F(x,y)=0\)在点\((x_0,y_0)\)的某一邻域内，恒能确定一个具有连续偏导数的函数\(y=f(x)\)，满足条件\(y_0=f(x_0)\)并有：	\[\frac{\mathop{}\!\mathrm{d}y}{\mathop{}\!\mathrm{d}x}=-\frac{F_x}{F_y}\]
设函数\(F(x,y,z)\)在点\(P(x_0,y_0,z_0)\)的某一邻域内具有连续偏导数，且\(F(x_0,y_0,z_0)=0,F_z(x_0,y_0,z_0)\not= 0\)，则方程\(F(x,y,z)=0\)在点\((x_0,y_0,z_0)\)的某一邻域内，恒能确定一个具有连续偏导数的函数\(z=f(x,y)\)，满足条件\(z_0=f(x_0,y_0)\)并有：	\[\frac{\partial z}{\partial x}=-\frac{F_x}{F_z},\frac{\partial z}{\partial y}=-\frac{F_y}{F_z}\]
设函数\(F(x,y,u,v),G(x,y,u,v)\)在点\(P(x_0,y_0,u_0,v_0)\)的某一邻域内具有对各个变量的连续偏导数，且\(F(x_0,y_0,u_0,v_0)=0,G(x_0,y_0,u_0,v_0)=0\)，雅克比行列式\(J\not= 0\)，则方程组\(F(x,y,u,v)=0,G(x,y,u,v)=0\)在点\((x_0,y_0,u_0,v_0)\)的某一邻域内，恒能确定一组具有连续偏导数的函数\(u=u(x,y),v=v(x,y)\)，满足条件\(u_0=u(x_0,y_0),v_0=v(x_0,y_0)\)并有：	\[J=\frac{\partial(F,G)}{\partial(u,v)}=\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v}\end{matrix}\right|\] \[\frac{\partial u}{\partial x}=-\frac 1J\frac{\partial(F,G)}{\partial(x,v)}=-\frac{\left|\begin{matrix}\frac{\partial F}{\partial x} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial x} & \frac{\partial G}{\partial v}\end{matrix}\right|}{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v}\end{matrix}\right|}\] \[\frac{\partial u}{\partial y}=-\frac 1J\frac{\partial(F,G)}{\partial(y,v)}=-\frac{\left|\begin{matrix}\frac{\partial F}{\partial y} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial y} & \frac{\partial G}{\partial v}\end{matrix}\right|}{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v}\end{matrix}\right|}\] \[\frac{\partial v}{\partial x}=-\frac 1J\frac{\partial(F,G)}{\partial(u,x)}=-\frac{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial x} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial x}\end{matrix}\right|}{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v}\end{matrix}\right|}\] \[\frac{\partial v}{\partial y}=-\frac 1J\frac{\partial(F,G)}{\partial(u,y)}=-\frac{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial y} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial y}\end{matrix}\right|}{\left|\begin{matrix}\frac{\partial F}{\partial u} & \frac{\partial F}{\partial v} \\ \frac{\partial G}{\partial u} & \frac{\partial G}{\partial v}\end{matrix}\right|}\]
全增量\(\Delta z=\)	\[f(x+\Delta x,y+\Delta y)-f(x,y)=A\Delta x+B\Delta y+o(\rho)\]
空间曲线\(x=\varphi(t),y=\psi(t),z=\omega(t)\)在\((x_0,y_0,z_0)\)的切向量	\[(\varphi'(t_0),\psi'(t_0),\omega'(t_0))\]
空间曲线\(x=\varphi(t),y=\psi(t),z=\omega(t)\)在\((x_0,y_0,z_0)\)的切线方程	\[\frac{x-x_0}{\varphi'(t_0)}=\frac{y-y_0}{\psi'(t_0)}=\frac{z-z_0}{\omega'(t_0)}\]
空间曲线\(x=\varphi(t),y=\psi(t),z=\omega(t)\)在\((x_0,y_0,z_0)\)的法平面方程	\[\varphi'(t_0)(x-x_0)+\psi'(t_0)(y-y_0)+\omega'(t_0)(z-z_0)=0\]
空间曲面\(F(x,y,z)=0\)在\((x_0,y_0,z_0)\)的法向量	\[(F_x(x_0,y_0,z_0),F_y(x_0,y_0,z_0),F_z(x_0,y_0,z_0))\]
空间曲面\(F(x,y,z)=0\)在\((x_0,y_0,z_0)\)的法线方程	\[\frac{x-x_0}{F_x(x_0,y_0,z_0)}=\frac{y-y_0}{F_y(x_0,y_0,z_0)}=\frac{z-z_0}{F_z(x_0,y_0,z_0)}\]
空间曲面\(F(x,y,z)=0\)在\((x_0,y_0,z_0)\)的切平面方程	\[F_x(x_0,y_0,z_0)(x-x_0)+F_y(x_0,y_0,z_0)(y-y_0)+F_z(x_0,y_0,z_0)(z-z_0)=0\]
方向导数\(\left.\frac{\partial f}{\partial l}\right|_{(x_0.y_0)}=\)	\[f_x(x_0,y_0)\cos\alpha+f_y(x_0,y_0)cos\beta=\nabla f(x_0,y_0)\cdot \hat e_l\]
梯度\(\nabla f(x_0,y_0)=\)	\[f_x(x_0,y_0)\hat i+f_y(x_0,y_0)\hat j\]
按定义求方向导数\(\left.\frac{\partial f}{\partial l}\right|_{P_0}=\)	\[\lim_{\rho\rightarrow 0}\frac{f(P)-f(P_0)}{\rho},\rho=|P_0P|\]
函数\(z=f(x,y)\)在\((x_0,y_0)\)具有偏导数的充分条件	\[f_x(x_0,y_0)=f_y(x_0,y_0)=0\]
\[\sum_{n=0}^{\infty}x^n=\]	\[\frac{1}{1-x},|x|<1\]
\[\sum_{n=1}^{\infty}x^n=\]	\[\frac{x}{1-x},|x|<1\]
函数展开成幂级数\[\frac{1}{1-x}=\]	\[\sum_{n=0}^{\infty}x^n\]
函数展开成幂级数\[\frac{1}{1+x}\]	\[\sum_{n=0}^{\infty}(-1)^nx^n\]
函数展开成幂级数\[e^x=\]	\[\sum_{n=0}^{\infty}\frac{x^n}{n!}\]
函数展开成幂级数\[\sin x=\]	\[\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+1}}{(2n+1)!}\]
函数展开成幂级数\[\cos x=\]	\[\sum_{n=0}^{\infty}(-1)^n\frac{x^{2n}}{(2n)!}\]
函数展开成幂级数\[\ln(1+x)=\]	\[\sum_{n=0}^{\infty}(-1)^n\frac{x^{n+1}}{n+1}\]
函数展开成幂级数\[(1+x)^a=\]	\[\sum_{n=0}^{\infty}\frac{a(a-1)\ldots(a-n+1)}{n!}x^n\]
两类曲线积分之间的联系	\[\begin{align*}\int_LP\mathop{}\!\mathrm{d}x+Q\mathop{}\!\mathrm{d}y=\int_L(P\cos\alpha+Q\cos\beta)\mathop{}\!\mathrm{d}s\\\int_\Gamma P\mathop{}\!\mathrm{d}x+Q\mathop{}\!\mathrm{d}y+R\mathop{}\!\mathrm{d}z=\int_\Gamma(P\cos\alpha+Q\cos\beta+R\cos\gamma)\mathop{}\!\mathrm{d}s\end{align*}\]
格林公式	\[\iint_D(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y})\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y=\oint_L P\mathop{}\!\mathrm{d}x+Q\mathop{}\!\mathrm{d}y\]
用格林公式将面积转化为曲线积分	\[A=\frac12\oint_Lx\mathop{}\!\mathrm{d}y-y\mathop{}\!\mathrm{d}x\]
两类曲面积分的关系	\[\iint_S P\mathop{}\!\mathrm{d}y\mathop{}\!\mathrm{d}z+Q\mathop{}\!\mathrm{d}z\mathop{}\!\mathrm{d}x+R\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y=\iint_S[P\cos\alpha+Q\cos\beta+R\cos\gamma]\mathop{}\!\mathrm{d}S\]
把组合式第二型曲面积分化为单一式第二型曲面积分	\[\iint_SP\mathop{}\!\mathrm{d}y\mathop{}\!\mathrm{d}z+Q\mathop{}\!\mathrm{d}z\mathop{}\!\mathrm{d}x+R\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y=\iint_S(-z_xP-z_yQ+R)\mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y\]
斯托克斯公式	\[\iint_S\left|\begin{matrix}\mathop{}\!\mathrm{d}y\mathop{}\!\mathrm{d}z & \mathop{}\!\mathrm{d}z\mathop{}\!\mathrm{d}x & \mathop{}\!\mathrm{d}x\mathop{}\!\mathrm{d}y \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R\end{matrix}\right|=\iint_S\left|\begin{matrix}\cos\alpha & \cos\beta & \cos\gamma \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R\end{matrix}\right|\mathop{}\!\mathrm{d}S=\oint_\Gamma P\mathop{}\!\mathrm{d}x+Q\mathop{}\!\mathrm{d}y+R\mathop{}\!\mathrm{d}z\]
散度\(\mathrm{div}\vec A=\)	\[\frac{\partial P}{\partial x}+\frac{\partial Q}{\partial y}+\frac{\partial R}{\partial z}\]
旋度\(\mathbf{rot}\vec A=\)	\[\left|\begin{matrix}\hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ P & Q & R\end{matrix}\right|\]
一般式空间曲线在点\(M\)处的切向量\(\vec n=\)	\[\left\{\left.\frac{\partial(F,G)}{\partial(y,z)}\right|_M,\left.\frac{\partial(F,G)}{\partial(z,x)}\right|_M,\left.\frac{\partial(F,G)}{\partial(x,y)}\right|_M\right\}\]