微积分复习(四)多元函数积分学

本文深入解析多重积分,涵盖二重积分、三重积分及其转换,包括直角坐标、极坐标和一般坐标变换,探讨线面积分及点函数积分,详解积分的性质与应用。

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二重积分

二重积分(f(x,y)f(x,y)f(x,y)σ\sigmaσ 上的黎曼积分)∬σf(x,y)dσ=lim⁡λ→0∑i=1nf(ξi,ηi)Δσi\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}=\displaystyle\lim_{\lambda \to 0}{\displaystyle\sum_{i=1}^{n}{f(\xi_i,\eta_i)\Delta \sigma_i}}σf(x,y)dσ=λ0limi=1nf(ξi,ηi)Δσi
绝对值不等式∣∬σf(x,y)dσ∣≤∬σ∣f(x,y)∣dσ\left|\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}\right| \leq \displaystyle\iint_{\sigma}{|f(x,y)|\mathrm{d}\sigma}σf(x,y)dσσf(x,y)dσ
二重积分中值定理 若 f(x,y)f(x,y)f(x,y) 在有界闭区域 σ\sigmaσ 上连续,则至少存在一点 P(x∗,y∗)∈σP(x^*,y^*) \in \sigmaP(x,y)σ,使得 ∬σf(x,y)dσ=f(x∗,y∗)σ\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}=f(x^*,y^*)\sigmaσf(x,y)dσ=f(x,y)σ,其中 f(x∗,y∗)=1σ∬σf(x,y)dσf(x^*,y^*)=\dfrac1\sigma\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}f(x,y)=σ1σf(x,y)dσ 称为 f(x,y)f(x,y)f(x,y)σ\sigmaσ 上的平均值。
推论 若 m≤f(x,y)≤Mm \leq f(x,y) \leq Mmf(x,y)M,则 mσ≤∬σf(x,y)dσ≤Mσm\sigma \leq \displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma} \leq M\sigmamσσf(x,y)dσMσ
二重积分转化为累次积分∬σf(x,y)dσ=∬σf(x,y)dxdy=∫abdx∫φ1(x)φ2(x)f(x,y)dy=∫cddy∫ψ1(y)ψ2(y)f(x,y)dx\begin{aligned}\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}&=\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}x\mathrm{d}y} \\ &=\displaystyle\int_{a}^{b}{\mathrm{d}x}\displaystyle\int_{\varphi_1(x)}^{\varphi_2(x)}{f(x,y)\mathrm{d}y}=\displaystyle\int_{c}^{d}{\mathrm{d}y}\displaystyle\int_{\psi_1(y)}^{\psi_2(y)}{f(x,y)\mathrm{d}x}\end{aligned}σf(x,y)dσ=σf(x,y)dxdy=abdxφ1(x)φ2(x)f(x,y)dy=cddyψ1(y)ψ2(y)f(x,y)dx二重积分的极坐标变换{x=rcos⁡θy=rsin⁡θ\begin{cases}x=r\cos\theta \\ y=r\sin\theta\end{cases}{x=rcosθy=rsinθ ∬σf(x,y)dσ=∬σf(rcos⁡θ,rsin⁡θ)rdrdθ=∫αβdθ∫r1(θ)r2(θ)f(rcos⁡θ,rsin⁡θ)rdr=∫r1r2dr∫θ1(r)θ2(r)f(rcos⁡θ,rsin⁡θ)rdθ\begin{aligned}\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}\sigma}&=\displaystyle\iint_{\sigma}{f(r\cos\theta,r\sin\theta)r\mathrm{d}r\mathrm{d}\theta} \\ &=\displaystyle\int_{\alpha}^{\beta}{\mathrm{d}\theta}\displaystyle\int_{r_1(\theta)}^{r_2(\theta)}{f(r\cos\theta,r\sin\theta)r\mathrm{d}r}=\displaystyle\int_{r_1}^{r_2}{\mathrm{d}r}\displaystyle\int_{\theta_1(r)}^{\theta_2(r)}{f(r\cos\theta,r\sin\theta)r\mathrm{d}\theta}\end{aligned}σf(x,y)dσ=σf(rcosθ,rsinθ)rdrdθ=αβdθr1(θ)r2(θ)f(rcosθ,rsinθ)rdr=r1r2drθ1(r)θ2(r)f(rcosθ,rsinθ)rdθ ⟸dσ=12(r+dr)2sin⁡dθ−12r2sin⁡dθ=rdrsin⁡dθ+12(dr)2sin⁡dθ∼rdrsin⁡dθ∼rdrdθ\Longleftarrow \mathrm{d}\sigma=\dfrac12(r+\mathrm{d}r)^2\sin\mathrm{d}\theta-\dfrac12r^2\sin\mathrm{d}\theta=r\mathrm{d}r\sin\mathrm{d}\theta+\dfrac12(\mathrm{d}r)^2\sin\mathrm{d}\theta \sim r\mathrm{d}r\sin\mathrm{d}\theta \sim r\mathrm{d}r\mathrm{d}\thetadσ=21(r+dr)2sindθ21r2sindθ=rdrsindθ+21(dr)2sindθrdrsindθrdrdθ
二重积分的一般坐标变换(雅可比行列式)∬σf(x,y)dxdy=∬σg(u,v)∣J∣dudv\displaystyle\iint_{\sigma}{f(x,y)\mathrm{d}x\mathrm{d}y}=\displaystyle\iint_{\sigma}{g(u,v)|J|\mathrm{d}u\mathrm{d}v}σf(x,y)dxdy=σg(u,v)JdudvJ=∂(x,y)∂(u,v)=∣∂x∂u∂x∂v∂y∂u∂y∂v∣=1∂(u,v)∂(x,y)=∣∂u∂x∂u∂y∂v∂x∂v∂y∣−1J=\dfrac{\partial(x,y)}{\partial(u,v)}=\begin{vmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v}\end{vmatrix}=\dfrac{1}{\dfrac{\partial(u,v)}{\partial(x,y)}}=\begin{vmatrix}\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y}\end{vmatrix}^{-1}J=(u,v)(x,y)=uxuyvxvy=(x,y)(u,v)1=xuxvyuyv1

三重积分

三重积分(密度函数 f(x,y,z)f(x,y,z)f(x,y,z) 在空间立体 VVV 上的质量)M=∭Vf(x,y,z)dV=lim⁡λ→0∑i=1nf(ξi,ηi,ζi)ΔViM=\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}V}=\displaystyle\lim_{\lambda \to 0}{\displaystyle\sum_{i=1}^{n}{f(\xi_i,\eta_i,\zeta_i)\Delta V_i}}M=Vf(x,y,z)dV=λ0limi=1nf(ξi,ηi,ζi)ΔVi
三重积分转化为累次积分(投影法)∭Vf(x,y,z)dV=∭Vf(x,y,z)dxdydz=∫abdx∫φ1(x)φ2(x)dy∫z1(x,y)z2(x,y)f(x,y,z)dz=∫cddy∫ψ1(y)ψ2(y)dx∫z1(x,y)z2(x,y)f(x,y,z)dz=∬σzxdσ∫y1(x,z)y2(x,z)f(x,y,z)dy=∬σyzdσ∫x1(y,z)x2(y,z)f(x,y,z)dx\begin{aligned}\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}V}&=\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z} \\ &=\displaystyle\int_{a}^{b}{\mathrm{d}x}\displaystyle\int_{\varphi_1(x)}^{\varphi_2(x)}{\mathrm{d}y}\displaystyle\int_{z_1(x,y)}^{z_2(x,y)}{f(x,y,z)\mathrm{d}z}=\displaystyle\int_{c}^{d}{\mathrm{d}y}\displaystyle\int_{\psi_1(y)}^{\psi_2(y)}{\mathrm{d}x}\displaystyle\int_{z_1(x,y)}^{z_2(x,y)}{f(x,y,z)\mathrm{d}z} \\ &=\displaystyle\iint_{\sigma_{zx}}{\mathrm{d}\sigma}\displaystyle\int_{y_1(x,z)}^{y_2(x,z)}{f(x,y,z)\mathrm{d}y} \\ &=\displaystyle\iint_{\sigma_{yz}}{\mathrm{d}\sigma}\displaystyle\int_{x_1(y,z)}^{x_2(y,z)}{f(x,y,z)\mathrm{d}x}\end{aligned}Vf(x,y,z)dV=Vf(x,y,z)dxdydz=abdxφ1(x)φ2(x)dyz1(x,y)z2(x,y)f(x,y,z)dz=cddyψ1(y)ψ2(y)dxz1(x,y)z2(x,y)f(x,y,z)dz=σzxdσy1(x,z)y2(x,z)f(x,y,z)dy=σyzdσx1(y,z)x2(y,z)f(x,y,z)dx三重积分转化为累次积分(截割法)∭Vf(x,y,z)dV=∭Vf(x,y,z)dxdydz=∫efdz∬Dzf(x,y,z)dσ=∫efdz∬Dzf(x,y,z)dxdy=∫cddy∬Dyf(x,y,z)dσ=∫abdx∬Dxf(x,y,z)dσ\begin{aligned}\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}V}&=\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z} \\ &=\displaystyle\int_{e}^{f}{\mathrm{d}z}\displaystyle\iint_{D_z}{f(x,y,z)\mathrm{d}\sigma}=\displaystyle\int_{e}^{f}{\mathrm{d}z}\displaystyle\iint_{D_z}{f(x,y,z)\mathrm{d}x\mathrm{d}y} \\ &=\displaystyle\int_{c}^{d}{\mathrm{d}y}\displaystyle\iint_{D_y}{f(x,y,z)\mathrm{d}\sigma} \\ &=\displaystyle\int_{a}^{b}{\mathrm{d}x}\displaystyle\iint_{D_x}{f(x,y,z)\mathrm{d}\sigma}\end{aligned}Vf(x,y,z)dV=Vf(x,y,z)dxdydz=efdzDzf(x,y,z)dσ=efdzDzf(x,y,z)dxdy=cddyDyf(x,y,z)dσ=abdxDxf(x,y,z)dσ三重积分的柱坐标变换{x=rcos⁡θy=rsin⁡θz=z\begin{cases}x=r\cos\theta \\ y=r\sin\theta \\ z=z\end{cases}x=rcosθy=rsinθz=z ∭Vf(x,y,z)dV=∭Vf(rcos⁡θ,rsin⁡θ,z)rdrdθdz=∬σrdrdθ∫z1(r,θ)z2(r,θ)f(rcos⁡θ,rsin⁡θ,z)dz\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}V}=\displaystyle\iiint_{V}{f(r\cos\theta,r\sin\theta,z)r\mathrm{d}r\mathrm{d}\theta\mathrm{d}z}=\displaystyle\iint_{\sigma}{r\mathrm{d}r\mathrm{d}\theta}\displaystyle\int_{z_1(r,\theta)}^{z_2(r,\theta)}{f(r\cos\theta,r\sin\theta,z)\mathrm{d}z}Vf(x,y,z)dV=Vf(rcosθ,rsinθ,z)rdrdθdz=σrdrdθz1(r,θ)z2(r,θ)f(rcosθ,rsinθ,z)dz三重积分的球坐标变换{x=ρsin⁡φcos⁡θy=ρsin⁡φsin⁡θz=ρcos⁡φ\begin{cases}x=\rho\sin\varphi\cos\theta \\ y=\rho\sin\varphi\sin\theta \\ z=\rho\cos\varphi\end{cases}x=ρsinφcosθy=ρsinφsinθz=ρcosφ ∭Vf(x,y,z)dV=∭Vf(ρsin⁡φcos⁡θ,ρsin⁡φsin⁡θ,ρcos⁡φ)ρ2sin⁡φdρdφdθ\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}V}=\displaystyle\iiint_{V}{f(\rho\sin\varphi\cos\theta,\rho\sin\varphi\sin\theta,\rho\cos\varphi)\rho^2\sin\varphi\mathrm{d}\rho\mathrm{d}\varphi\mathrm{d}\theta}Vf(x,y,z)dV=Vf(ρsinφcosθ,ρsinφsinθ,ρcosφ)ρ2sinφdρdφdθ ⟸dV=ρtan⁡dφ⋅ρsin⁡φtan⁡dθ⋅dρ=ρ2sin⁡φdρtan⁡dφtan⁡dθ∼ρ2sin⁡φdρdφdθ\Longleftarrow \mathrm{d}V=\rho\tan\mathrm{d}\varphi \cdot \rho\sin\varphi\tan\mathrm{d}\theta \cdot \mathrm{d}\rho=\rho^2\sin\varphi\mathrm{d}\rho\tan\mathrm{d}\varphi\tan\mathrm{d}\theta \sim \rho^2\sin\varphi\mathrm{d}\rho\mathrm{d}\varphi\mathrm{d}\thetadV=ρtandφρsinφtandθdρ=ρ2sinφdρtandφtandθρ2sinφdρdφdθ
三重积分的一般坐标变换(雅可比行列式)∭Vf(x,y,z)dxdydz=∭Vg(u,v,w)∣J∣dudvdw\displaystyle\iiint_{V}{f(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z}=\displaystyle\iiint_{V}{g(u,v,w)|J|\mathrm{d}u\mathrm{d}v\mathrm{d}w}Vf(x,y,z)dxdydz=Vg(u,v,w)JdudvdwJ=∂(x,y,z)∂(u,v,w)=∣∂x∂u∂x∂v∂x∂w∂y∂u∂y∂v∂y∂w∂z∂u∂z∂v∂z∂w∣=1∂(u,v,w)∂(x,y,z)=∣∂u∂x∂u∂y∂u∂z∂v∂x∂v∂y∂v∂z∂w∂x∂w∂y∂w∂z∣−1J=\dfrac{\partial(x,y,z)}{\partial(u,v,w)}=\begin{vmatrix}\dfrac{\partial x}{\partial u} & \dfrac{\partial x}{\partial v} & \dfrac{\partial x}{\partial w} \\ \dfrac{\partial y}{\partial u} & \dfrac{\partial y}{\partial v} & \dfrac{\partial y}{\partial w} \\ \dfrac{\partial z}{\partial u} & \dfrac{\partial z}{\partial v} & \dfrac{\partial z}{\partial w}\end{vmatrix}=\dfrac{1}{\dfrac{\partial(u,v,w)}{\partial(x,y,z)}}=\begin{vmatrix}\dfrac{\partial u}{\partial x} & \dfrac{\partial u}{\partial y} & \dfrac{\partial u}{\partial z} \\ \dfrac{\partial v}{\partial x} & \dfrac{\partial v}{\partial y} & \dfrac{\partial v}{\partial z} \\ \dfrac{\partial w}{\partial x} & \dfrac{\partial w}{\partial y} & \dfrac{\partial w}{\partial z}\end{vmatrix}^{-1}J=(u,v,w)(x,y,z)=uxuyuzvxvyvzwxwywz=(x,y,z)(u,v,w)1=xuxvxwyuyvywzuzvzw1

第一类线面积分

第一类曲线积分(弧长积分)∫Γf(P)ds=lim⁡λ→0∑i=1nf(Pi)Δsi=∫αβf(x(t),y(t),z(t))x′2(t)+y′2(t)+z′2(t)dt\displaystyle\int_{\Gamma}{f(P)\mathrm{d}s}=\displaystyle\lim_{\lambda \to 0}{\displaystyle\sum_{i=1}^{n}{f(P_i)\Delta s_i}}=\displaystyle\int_{\alpha}^{\beta}{f(x(t),y(t),z(t))\sqrt{x'^2(t)+y'^2(t)+z'^2(t)}\mathrm{d}t}Γf(P)ds=λ0limi=1nf(Pi)Δsi=αβf(x(t),y(t),z(t))x2(t)+y2(t)+z2(t)dt ⟸ds=[x(t+dt)−x(t)]2+[y(t+dt)−y(t)]2+[z(t+dt)−z(t)]2=x′2(ξ1)+y′2(ξ2)+z′2(ξ3)dt (t<ξ1,ξ2,ξ3<t+dt)=x′2(t)+y′2(t)+z′2(t)dt\begin{aligned}\Longleftarrow \mathrm{d}s&=\sqrt{[x(t+\mathrm{d}t)-x(t)]^2+[y(t+\mathrm{d}t)-y(t)]^2+[z(t+\mathrm{d}t)-z(t)]^2} \\ &=\sqrt{x'^2(\xi_1)+y'^2(\xi_2)+z'^2(\xi_3)}\mathrm{d}t \ (t<\xi_1,\xi_2,\xi_3<t+\mathrm{d}t) \\ &=\sqrt{x'^2(t)+y'^2(t)+z'^2(t)}\mathrm{d}t\end{aligned}ds=[x(t+dt)x(t)]2+[y(t+dt)y(t)]2+[z(t+dt)z(t)]2=x2(ξ1)+y2(ξ2)+z2(ξ3)dt (t<ξ1,ξ2,ξ3<t+dt)=x2(t)+y2(t)+z2(t)dt
平面曲线弧长积分∫Γf(x,y)ds=∫αβf(x(t),y(t))x′2(t)+y′2(t)dt\displaystyle\int_{\Gamma}{f(x,y)\mathrm{d}s}=\displaystyle\int_{\alpha}^{\beta}{f(x(t),y(t))\sqrt{x'^2(t)+y'^2(t)}\mathrm{d}t}Γf(x,y)ds=αβf(x(t),y(t))x2(t)+y2(t)dt
(1) Γ:y=φ(x),x∈[a,b]⟹∫Γf(x,y)ds=∫abf(x,φ(x))1+φ′2(x)dx\Gamma:y=\varphi(x),x \in [a,b] \Longrightarrow \displaystyle\int_{\Gamma}{f(x,y)\mathrm{d}s}=\displaystyle\int_{a}^{b}{f(x,\varphi(x))\sqrt{1+\varphi'^2(x)}\mathrm{d}x}Γ:y=φ(x),x[a,b]Γf(x,y)ds=abf(x,φ(x))1+φ2(x)dx
(2) Γ:x=ψ(y),y∈[c,d]⟹∫Γf(x,y)ds=∫cdf(ψ(y),y)1+ψ′2(y)dy\Gamma:x=\psi(y),y \in [c,d] \Longrightarrow \displaystyle\int_{\Gamma}{f(x,y)\mathrm{d}s}=\displaystyle\int_{c}^{d}{f(\psi(y),y)\sqrt{1+\psi'^2(y)}\mathrm{d}y}Γ:x=ψ(y),y[c,d]Γf(x,y)ds=cdf(ψ(y),y)1+ψ2(y)dy
(3) Γ:r=r(θ),θ∈[α,β]⟹∫Γf(x,y)ds=∫αβf(rcos⁡θ,rsin⁡θ)r2(θ)+r′2(θ)dθ\Gamma:r=r(\theta),\theta \in [\alpha,\beta] \Longrightarrow \displaystyle\int_{\Gamma}{f(x,y)\mathrm{d}s}=\displaystyle\int_{\alpha}^{\beta}{f(r\cos\theta,r\sin\theta)\sqrt{r^2(\theta)+r'^2(\theta)}\mathrm{d}\theta}Γ:r=r(θ),θ[α,β]Γf(x,y)ds=αβf(rcosθ,rsinθ)r2(θ)+r2(θ)dθ
第一类曲面积分∬Sf(P)dS=lim⁡λ→0∑i=1nf(Pi)ΔSi\displaystyle\iint_{S}{f(P)\mathrm{d}S}=\displaystyle\lim_{\lambda \to 0}{\displaystyle\sum_{i=1}^{n}{f(P_i)\Delta S_i}}Sf(P)dS=λ0limi=1nf(Pi)ΔSi
(1) S:z=z(x,y),(x,y)∈σxy⟹∬Sf(x,y,z)dS=∬σxyf(x,y,z(x,y))1+zx′2+zy′2dσS:z=z(x,y),(x,y) \in \sigma_{xy} \Longrightarrow \displaystyle\iint_{S}{f(x,y,z)\mathrm{d}S}=\displaystyle\iint_{\sigma_{xy}}{f(x,y,z(x,y))\sqrt{1+z_x'^2+z_y'^2}}\mathrm{d}\sigmaS:z=z(x,y),(x,y)σxySf(x,y,z)dS=σxyf(x,y,z(x,y))1+zx2+zy2dσ
  取面微元 dS\mathrm{d}SdS 上一点 P(x,y,z(x,y))P(x,y,z(x,y))P(x,y,z(x,y)),则该点处法线的方向矢量 n=±{zx′,zy′,−1}\boldsymbol{n}=\pm \{z_x',z_y',-1\}n=±{zx,zy,1},记 γ\gammaγn\boldsymbol{n}nzzz 轴正方向的夹角,有 cos⁡γ=±11+zx′2+zy′2\cos\gamma=\pm \dfrac{1}{\sqrt{1+z_x'^2+z_y'^2}}cosγ=±1+zx2+zy21,故 dS\mathrm{d}SdSOxyOxyOxy 平面上的投影面积 dσ=∣cos⁡γ∣⋅dS\mathrm{d}\sigma=|\cos\gamma| \cdot \mathrm{d}Sdσ=cosγdS,可得 dS=dσ∣cos⁡γ∣=1+zx′2+zy′2dσ\mathrm{d}S=\dfrac{\mathrm{d}\sigma}{|\cos\gamma|}=\sqrt{1+z_x'^2+z_y'^2}\mathrm{d}\sigmadS=cosγdσ=1+zx2+zy2dσ
(2) S:y=y(x,z),(x,z)∈σzx⟹∬Sf(x,y,z)dS=∬σzxf(x,y(x,z),z)1+yx′2+yz′2dσS:y=y(x,z),(x,z) \in \sigma_{zx} \Longrightarrow \displaystyle\iint_{S}{f(x,y,z)\mathrm{d}S}=\displaystyle\iint_{\sigma_{zx}}{f(x,y(x,z),z)\sqrt{1+y_x'^2+y_z'^2}}\mathrm{d}\sigmaS:y=y(x,z),(x,z)σzxSf(x,y,z)dS=σzxf(x,y(x,z),z)1+yx2+yz2dσ
(3) S:x=x(y,z),(y,z)∈σyz⟹∬Sf(x,y,z)dS=∬σyzf(x(y,z),y,z)1+xy′2+xz′2dσS:x=x(y,z),(y,z) \in \sigma_{yz} \Longrightarrow \displaystyle\iint_{S}{f(x,y,z)\mathrm{d}S}=\displaystyle\iint_{\sigma_{yz}}{f(x(y,z),y,z)\sqrt{1+x_y'^2+x_z'^2}}\mathrm{d}\sigmaS:x=x(y,z),(y,z)σyzSf(x,y,z)dS=σyzf(x(y,z),y,z)1+xy2+xz2dσ
(4)S:F(x,y,z)=0S:F(x,y,z)=0S:F(x,y,z)=0 确定隐函数 z=z(x,y),(x,y)∈σxyz=z(x,y),(x,y) \in \sigma_{xy}z=z(x,y),(x,y)σxy,且 ∂z∂x=−Fx′Fz′\dfrac{\partial z}{\partial x}=-\dfrac{F_x'}{F_z'}xz=FzFx∂z∂y=−Fy′Fz′\dfrac{\partial z}{\partial y}=-\dfrac{F_y'}{F_z'}yz=FzFy 连续,则 ∬Sf(x,y,z)dS=∬σxyf(x,y,z(x,y))Fx′2+Fy′2+Fz′2∣Fz′∣dσ\displaystyle\iint_{S}{f(x,y,z)\mathrm{d}S}=\displaystyle\iint_{\sigma_{xy}}{f(x,y,z(x,y))\dfrac{\sqrt{F_x'^2+F_y'^2+F_z'^2}}{|F_z'|}\mathrm{d}\sigma}Sf(x,y,z)dS=σxyf(x,y,z(x,y))FzFx2+Fy2+Fz2dσ

点函数积分

点函数积分∫Ωf(P)dΩ=lim⁡λ→0∑i=1nf(Pi)ΔΩi\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\lim_{\lambda \to 0}{\displaystyle\sum_{i=1}^{n}{f(P_i)\Delta \Omega_i}}Ωf(P)dΩ=λ0limi=1nf(Pi)ΔΩi
点函数积分保号性 若 f(P)≤g(P),P∈Ωf(P) \leq g(P),P \in \Omegaf(P)g(P),PΩ,则 ∫Ωf(P)dΩ≤∫Ωg(P)dΩ\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega} \leq \displaystyle\int_{\Omega}{g(P)\mathrm{d}\Omega}Ωf(P)dΩΩg(P)dΩ;若连续函数 f(P)≤g(P),P∈Ωf(P) \leq g(P),P \in \Omegaf(P)g(P),PΩf(P)≢g(P)f(P) \not\equiv g(P)f(P)g(P),则 ∫Ωf(P)dΩ<∫Ωg(P)dΩ\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}<\displaystyle\int_{\Omega}{g(P)\mathrm{d}\Omega}Ωf(P)dΩ<Ωg(P)dΩ
绝对值不等式∣∫Ωf(P)dΩ∣≤∫Ω∣f(P)∣dΩ\left|\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}\right| \leq \displaystyle\int_{\Omega}{|f(P)|\mathrm{d}\Omega}Ωf(P)dΩΩf(P)dΩ
点函数积分中值定理 若 f(P)f(P)f(P) 在有界闭区域 Ω\OmegaΩ 上连续,则至少存在一点 P∗∈ΩP^* \in \OmegaPΩ,使得 ∫Ωf(P)dΩ=f(P∗)Ω\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=f(P^*)\OmegaΩf(P)dΩ=f(P)Ω,其中 f(P∗)=1Ω∫Ωf(P)dΩf(P^*)=\dfrac1\Omega\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}f(P)=Ω1Ωf(P)dΩ 称为 f(P)f(P)f(P)Ω\OmegaΩ 上的平均值。
推论 若 m≤f(P)≤Mm \leq f(P) \leq Mmf(P)M,则 mΩ≤∫Ωf(P)dΩ≤MΩm\Omega \leq \displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega} \leq M\OmegamΩΩf(P)dΩMΩ
点函数的分类
(1)Ω=[a,b]⊂R\Omega=[a,b] \subset \mathbb{R}Ω=[a,b]R,此时 f(P)=f(x)f(P)=f(x)f(P)=f(x),则 ∫Ωf(P)dΩ=∫abf(x)dx\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\int_a^b{f(x)\mathrm{d}x}Ωf(P)dΩ=abf(x)dx(一元函数定积分);
(2)Ω=s⊂R2\Omega=s \subset \mathbb{R}^2Ω=sR2sss 是平面曲线,此时 f(P)=f(x,y)f(P)=f(x,y)f(P)=f(x,y),则 ∫Ωf(P)dΩ=∫sf(x,y)ds\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\int_s{f(x,y)\mathrm{d}s}Ωf(P)dΩ=sf(x,y)ds(平面弧长积分);
(3)Ω=s⊂R3\Omega=s \subset \mathbb{R}^3Ω=sR3sss 是空间曲线,此时 f(P)=f(x,y,z)f(P)=f(x,y,z)f(P)=f(x,y,z),则 ∫Ωf(P)dΩ=∫sf(x,y,z)ds\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\int_s{f(x,y,z)\mathrm{d}s}Ωf(P)dΩ=sf(x,y,z)ds(空间弧长积分);
(4)Ω=σ⊂R2\Omega=\sigma \subset \mathbb{R}^2Ω=σR2σ\sigmaσ 是平面区域,此时 f(P)=f(x,y)f(P)=f(x,y)f(P)=f(x,y),则 ∫Ωf(P)dΩ=∬σf(x,y)dσ\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\iint_\sigma{f(x,y)\mathrm{d}\sigma}Ωf(P)dΩ=σf(x,y)dσ(二重积分);
(5)Ω=S⊂R3\Omega=S \subset \mathbb{R}^3Ω=SR3SSS 是空间曲面,此时 f(P)=f(x,y,z)f(P)=f(x,y,z)f(P)=f(x,y,z),则 ∫Ωf(P)dΩ=∬Sf(x,y,z)dS\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\iint_S{f(x,y,z)\mathrm{d}S}Ωf(P)dΩ=Sf(x,y,z)dS(第一类曲面积分);
(6)Ω=V⊂R3\Omega=V \subset \mathbb{R}^3Ω=VR3VVV 是空间立体,此时 f(P)=f(x,y,z)f(P)=f(x,y,z)f(P)=f(x,y,z),则 ∫Ωf(P)dΩ=∭Vf(x,y,z)dV\displaystyle\int_{\Omega}{f(P)\mathrm{d}\Omega}=\displaystyle\iiint_V{f(x,y,z)\mathrm{d}V}Ωf(P)dΩ=Vf(x,y,z)dV(三重积分)。
重心{xˉ=1M∫Ωμ(P)xdΩyˉ=1M∫Ωμ(P)ydΩzˉ=1M∫Ωμ(P)zdΩ\begin{cases}\bar{x}=\dfrac1M\displaystyle\int_{\Omega}{\mu(P)x\mathrm{d}\Omega} \\ \bar{y}=\dfrac1M\displaystyle\int_{\Omega}{\mu(P)y\mathrm{d}\Omega} \\ \bar{z}=\dfrac1M\displaystyle\int_{\Omega}{\mu(P)z\mathrm{d}\Omega}\end{cases}xˉ=M1Ωμ(P)xdΩyˉ=M1Ωμ(P)ydΩzˉ=M1Ωμ(P)zdΩ
转动惯量(1) Ω⊂R3⟹{Iz=∫Ω(x2+y2)μ(P)dΩIy=∫Ω(z2+x2)μ(P)dΩIx=∫Ω(y2+z2)μ(P)dΩ\Omega \subset \mathbb{R}^3 \Longrightarrow \begin{cases}I_z=\displaystyle\int_{\Omega}{(x^2+y^2)\mu(P)\mathrm{d}\Omega} \\ I_y=\displaystyle\int_{\Omega}{(z^2+x^2)\mu(P)\mathrm{d}\Omega} \\ I_x=\displaystyle\int_{\Omega}{(y^2+z^2)\mu(P)\mathrm{d}\Omega}\end{cases}ΩR3Iz=Ω(x2+y2)μ(P)dΩIy=Ω(z2+x2)μ(P)dΩIx=Ω(y2+z2)μ(P)dΩ (2) Ω⊂R2⟹{Ix=∫Ωy2μ(P)dΩIy=∫Ωx2μ(P)dΩ\Omega \subset \mathbb{R}^2 \Longrightarrow \begin{cases}I_x=\displaystyle\int_{\Omega}{y^2\mu(P)\mathrm{d}\Omega} \\ I_y=\displaystyle\int_{\Omega}{x^2\mu(P)\mathrm{d}\Omega}\end{cases}ΩR2Ix=Ωy2μ(P)dΩIy=Ωx2μ(P)dΩ
引力{Fx=km∫Ωμ(P)(x−x0)r3dΩFy=km∫Ωμ(P)(y−y0)r3dΩFz=km∫Ωμ(P)(z−z0)r3dΩ , r=(x−x0)2+(y−y0)2+(z−z0)2\begin{cases}F_x=km\displaystyle\int_{\Omega}{\dfrac{\mu(P)(x-x_0)}{r^3}\mathrm{d}\Omega} \\ F_y=km\displaystyle\int_{\Omega}{\dfrac{\mu(P)(y-y_0)}{r^3}\mathrm{d}\Omega} \\ F_z=km\displaystyle\int_{\Omega}{\dfrac{\mu(P)(z-z_0)}{r^3}\mathrm{d}\Omega}\end{cases} \ , \ r=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}Fx=kmΩr3μ(P)(xx0)dΩFy=kmΩr3μ(P)(yy0)dΩFz=kmΩr3μ(P)(zz0)dΩ , r=(xx0)2+(yy0)2+(zz0)2

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