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一型线面积分

一型曲线积分​

光滑曲线质量，弧微分ds=[x′(t)]2+[y′(t)]2+[z′(t)]2dt\mathrm{ds}=\sqrt{[x'(t)]^2+[y'(t)]^2+[z'(t)]^2}\mathrm{dt}

性质公式
曲线弧长∫Γ1ds=lΓ\int_{\Gamma}1ds=l_{\Gamma}
可积有界∫Γf(x,y,z)ds=A  ⟹  m≤f(x,y,z)≤M\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}=A\implies m\le f\left(x,y,z\right)\le M
保号性∥∫Γf(x,y,z)ds∥≤∬Γ∥f(x,y,z)∥ds\left\|\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}\right\|\le\iint_{\Gamma_{}}\left\|f\left(x,y,z\right)\right\|\mathrm{ds}
估值定理mlΓ≤∫Γf(x,y,z)ds≤MlΓml_\Gamma\le\int_{\Gamma}f\left(x,y,z\right)ds\le Ml_\Gamma
线性可拆∫Γ[k1f(x,y,z)+k2g(x,y,z)]ds=k1∫Γf(x,y,z)ds+k2∫Γg(x,y,z)ds\int_{\Gamma}\left\lbrack k_1f\left(x,y,z\right)+k_2g\left(x,y,z\right)]\mathrm{ds}\right.=k_1\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}+k_2\int_{\Gamma}g\left(x,y,z\right)\mathrm{ds}
区域可加∫Γf(x,y,z)ds=∫Γ1f(x,y,z)ds+∫Γ2f(x,y,z)ds\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}=\int_{\Gamma_1}f\left(x,y,z\right)\mathrm{ds}+\int_{\Gamma_2}f\left(x,y,z\right)\mathrm{ds}
中值定理∫Γf(x,y,z)ds=f(ξ,η,ζ)lΓ\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}=f\left(\xi,\eta,\zeta\right)l_{\Gamma}
普通对称∫Γf(x,y,z)ds={2∫Γ1f(x,y,z)dsf(−x,y,z)=f(x,y,z)0,f(x,y,z)=−f(−x,y,z)\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}=\left\{\begin{array}{l}2\int_{\Gamma_1}f\left(x,y,z\right)\mathrm{ds}& f\left(-x,y,z\right)=f\left(x,y,z\right)\\ 0,& f(x,y,z)=-f(-x,y,z)\end{array}\right.
轮换对称∫Γf(x,y,z)ds=∫Γf(y,x,z)ds\int_{\Gamma}f\left(x,y,z\right)\mathrm{ds}= \int_{\Gamma}f\left(y,x,z\right)\mathrm{ds}

将边界方程带入被积函数可化简

一型曲面积分​

光滑曲面薄片质量。dS=1+[zx′]2+[zy′]2dxdy\mathrm{dS}=\sqrt{1+[z'_x]^2+[z'_y]^2}\mathrm{dxdy}，基本计算方法：

∬Σf(x,y,z)dS=∬f[x,y,z(x,y)]1+(zx′)2+(zy′)2dxdy\iint_{\Sigma}f\left(x,y,z\right)dS=\iint f\left\lbrack x,y,z\left(x,y\right)\right\rbrack\sqrt{1+\left(z_{x}^{\prime}\right)^2+\left(z_{y}^{\prime}\right)^2}\mathrm{dxdy}

性质公式
曲面面积∬Σ1dS=S\iint_{\Sigma}1dS=S
可积有界∬Σf(x,y,z)dS=A  ⟹  m≤f(x,y,z)≤M\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}=A\implies m\le f\left(x,y,z\right)\le M
保号性∥∬Σf(x,y,z)dS∥≤∬Σ∥f(x,y,z)∥dS\left\|\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}\right\|\le\iint_{\Sigma_{}}\left\|f\left(x,y,z\right)\right\|\mathrm{dS}
估值定理mS≤∬Σf(x,y,z)dS≤MSmS\le\iint_{\Sigma}f\left(x,y,z\right)dS\le MS
线性可拆∬Σ[k1f(x,y,z)+k2g(x,y,z)]dS=k1∬Σf(x,y,z)dS+k2∬Σg(x,y,z)dS\iint_{\Sigma}\left\lbrack k_1f\left(x,y,z\right)+k_2g\left(x,y,z\right)]\mathrm{dS}\right.=k_1\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}+k_2\iint_{\Sigma}g\left(x,y,z\right)\mathrm{dS}
区域可加∬Σf(x,y,z)dS=∬Σ1f(x,y,z)dS+∬Σ2f(x,y,z)dS\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}=\iint_{\Sigma_1}f\left(x,y,z\right)\mathrm{dS}+\iint_{\Sigma_2}f\left(x,y,z\right)\mathrm{dS}
中值定理∬Σf(x,y,z)dS=f(ξ,η,ζ)S\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}=f\left(\xi,\eta,\zeta\right)S
普通对称∬Σf(x,y,z)dS={2∬Σ1f(x,y,z)dSf(−x,y,z)=f(x,y,z)0,f(x,y,z)=−f(−x,y,z)\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}=\left\{\begin{array}{l}2\iint_{\Sigma_1}f\left(x,y,z\right)\mathrm{dS}\\ f\left(-x,y,z\right)=f\left(x,y,z\right)\\ 0,\\ f(x,y,z)=-f(-x,y,z)\end{array}\right.
轮换对称∬Σf(x,y,z)dS=∬Σf(y,x,z)dS\iint_{\Sigma}f\left(x,y,z\right)\mathrm{dS}= \iint_{\Sigma}f\left(y,x,z\right)\mathrm{dS}

将边界方程带入被积函数可化简

一型积分的应用​

质心=重心，当ρ\rho是常数时，质心=形心。xˉ=∫xρ  dV∫ρdV;yˉ=∫yρ  dV∫ρdV;zˉ=∫zρ  dV∫ρdV\bar{x}=\dfrac{\int x\rho\;\mathrm{dV}}{\int\rho\mathrm{dV}};\bar{y}=\dfrac{\int y\rho\;\mathrm{dV}}{\int\rho\mathrm{dV}};\bar{z}=\dfrac{\int z\rho\;\mathrm{dV}}{\int\rho\mathrm{dV}}

转动惯量：Ix=∬Dy2ρ(x,y)dσ=∭Ω(y2+z2)ρ(x,y,z)dvI_{x}=\iint_{D}y^2\rho\left(x,y\right)d\sigma = \iiint _\Omega(y^2+z^2)\rho(x,y,z)dv

对于一个质点的转动惯量：I=mr2I=mr^2

引力Fx=Gm∭Ωρ(x,y,z)(x−x0)[(x−x0)2+(y−y0)2+(z+z0)2]32dvF_x=Gm\iiint_\Omega \dfrac{\rho (x,y,z)(x-x_0)}{[(x-x_0)^2+(y-y_0)^2+(z+z_0)^2]^{\frac{3}{2}}}dv

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