第二类曲线、曲面积分计算公式

下面将列出常用正交坐标系下的第二类曲线、曲面积分的直接计算公式。以下默认被积函数为对应正交坐标系下形如 f ⃗ = ( P , Q , R ) \vec{f}=(P,Q,R) f ​=(P,Q,R) 的矢量函数。

第二类曲线积分是在有向曲线的弧长上对矢量函数进行积分。

假定积分区域为 Γ : { x = x ( s ) y = y ( s ) z = z ( s ) ( s ∈ ( s ‾ , s ‾ ) ) \Gamma:\left\{\begin{array}{l} x=x(s)\\ y=y(s)\\ z=z(s) \end{array}\right.\\ (s\in(\underline{s},\overline{s})) Γ:⎩⎨⎧​x=x(s)y=y(s)z=z(s)​(s∈(s​,s)) 并规定 ( x s ′ , y s ′ , z s ′ ) (x_s',y_s',z_s') (xs′​,ys′​,zs′​) 的方向为正方向，则有 ∫ Γ f ⃗ ( x , y , z ) ⋅ d l ⃗ = ∫ Γ P d x + Q d y + R d z = ∫ s ‾ s ‾ ( P x s ′ + Q y s ′ + R z s ′ ) d s \int_{\Gamma}\vec{f}(x,y,z)\cdot\vec{\mathrm{d}l}=\int_{\Gamma}P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z\\ =\int_{\underline{s}}^{\overline{s}}(Px_s'+Qy_s'+Rz_s')\mathrm{d}s ∫Γ​f ​(x,y,z)⋅dl =∫Γ​Pdx+Qdy+Rdz=∫s​s​(Pxs′​+Qys′​+Rzs′​)ds 特殊地，若积分区域可以表示为 Γ : { x = x ( z ) y = y ( z ) z = z ( z ∈ ( z ‾ , z ‾ ) ) \Gamma:\left\{\begin{array}{l} x=x(z)\\ y=y(z)\\ z=z \end{array}\right.\\ (z\in(\underline{z},\overline{z})) Γ:⎩⎨⎧​x=x(z)y=y(z)z=z​(z∈(z​,z)) 并规定 ( x z ′ , y z ′ , 1 ) (x_z',y_z',1) (xz′​,yz′​,1) 的方向为正方向，则有 ∫ Γ f ⃗ ( x , y , z ) ⋅ d l ⃗ = ∫ Γ P d x + Q d y + R d z = ∫ z ‾ z ‾ ( P x z ′ + Q y z ′ + R ) d z \int_{\Gamma}\vec{f}(x,y,z)\cdot\vec{\mathrm{d}l}=\int_{\Gamma}P\mathrm{d}x+Q\mathrm{d}y+R\mathrm{d}z\\ =\int_{\underline{z}}^{\overline{z}}(Px_z'+Qy_z'+R)\mathrm{d}z ∫Γ​f ​(x,y,z)⋅dl =∫Γ​Pdx+Qdy+Rdz=∫z​z​(Pxz′​+Qyz′​+R)dz

假定积分区域为 Γ : { r = r ( s ) θ = θ ( s ) ϕ = ϕ ( s ) ( s ∈ ( s ‾ , s ‾ ) ) \Gamma:\left\{\begin{array}{l} r=r(s)\\ \theta=\theta(s)\\ \phi=\phi(s) \end{array}\right.\\ (s\in(\underline{s},\overline{s})) Γ:⎩⎨⎧​r=r(s)θ=θ(s)ϕ=ϕ(s)​(s∈(s​,s)) 并规定 ( r s ′ , θ s ′ , ϕ s ′ ) (r_s',\theta_s',\phi_s') (rs′​,θs′​,ϕs′​) 的方向为正方向，则有 ∫ Γ f ⃗ ( r , θ , ϕ ) ⋅ d l ⃗ = ∫ Γ P d r + Q r d θ + R r s i n θ d ϕ = ∫ s ‾ s ‾ ( P r s ′ + Q r θ s ′ + R r s i n θ ϕ s ′ ) d s \int_{\Gamma}\vec{f}(r,\theta,\phi)\cdot\vec{\mathrm{d}l}=\int_{\Gamma}P\mathrm{d}r+Qr\mathrm{d}\theta+Rr\mathrm{sin}\theta\mathrm{d}\phi\\ =\int_{\underline{s}}^{\overline{s}}(Pr_s'+Qr\theta_s'+Rr\mathrm{sin}\theta\phi_s')\mathrm{d}s ∫Γ​f ​(r,θ,ϕ)⋅dl =∫Γ​Pdr+Qrdθ+Rrsinθdϕ=∫s​s​(Prs′​+Qrθs′​+Rrsinθϕs′​)ds

假定积分区域为 Γ : { r = r ( s ) ϕ = ϕ ( s ) z = z ( s ) ( s ∈ ( s ‾ , s ‾ ) ) \Gamma:\left\{\begin{array}{l} r=r(s)\\ \phi=\phi(s)\\ z=z(s) \end{array}\right.\\ (s\in(\underline{s},\overline{s})) Γ:⎩⎨⎧​r=r(s)ϕ=ϕ(s)z=z(s)​(s∈(s​,s)) 并规定 ( r s ′ , ϕ s ′ , z s ′ ) (r_s',\phi_s',z_s') (rs′​,ϕs′​,zs′​) 的方向为正方向，则有 ∫ Γ f ⃗ ( r , ϕ , z ) ⋅ d l ⃗ = ∫ Γ P d r + Q r d ϕ + R d z = ∫ s ‾ s ‾ ( P r s ′ + Q r ϕ s ′ + R z s ′ ) d s \int_{\Gamma}\vec{f}(r,\phi,z)\cdot\vec{\mathrm{d}l}=\int_{\Gamma}P\mathrm{d}r+Qr\mathrm{d}\phi+R\mathrm{d}z\\ =\int_{\underline{s}}^{\overline{s}}(Pr_s'+Qr\phi_s'+Rz_s')\mathrm{d}s ∫Γ​f ​(r,ϕ,z)⋅dl =∫Γ​Pdr+Qrdϕ+Rdz=∫s​s​(Prs′​+Qrϕs′​+Rzs′​)ds

第二类曲面积分是在有向曲面的面积上对矢量函数进行积分。

假定积分区域为 Σ : { x = x ( s , t ) y = y ( s , t ) z = z ( s , t ) ( ( s , t ) ∈ Σ ′ ) \Sigma:\left\{\begin{array}{l} x=x(s,t)\\ y=y(s,t)\\ z=z(s,t) \end{array}\right.\\ ((s,t)\in\Sigma') Σ:⎩⎨⎧​x=x(s,t)y=y(s,t)z=z(s,t)​((s,t)∈Σ′) 并规定 ( x s ′ , y s ′ , z s ′ ) × ( x t ′ , y t ′ , z t ′ ) (x_s',y_s',z_s')\times(x_t',y_t',z_t') (xs′​,ys′​,zs′​)×(xt′​,yt′​,zt′​) 的方向为正方向，则有 ∬ Σ f ⃗ ( x , y , z ) ⋅ d σ ⃗ = ∬ Σ P d y ∧ d z + Q d z ∧ d x + R d x ∧ d y = ∬ Σ ′ ( P , Q , R ) ⋅ [ ( x s ′ , y s ′ , z s ′ ) × ( x t ′ , y t ′ , z t ′ ) ] d s d t = ∬ Σ ′ [ P ( y s ′ z t ′ − z s ′ y t ′ ) + Q ( z s ′ x t ′ − x s ′ z t ′ ) + R ( x s ′ y t ′ − y s ′ x t ′ ) ] d s d t \iint_{\Sigma}\vec{f}(x,y,z)\cdot\vec{\mathrm{d}\sigma}=\iint_{\Sigma}P\mathrm{d}y\wedge\mathrm{d}z+Q\mathrm{d}z\wedge\mathrm{d}x+R\mathrm{d}x\wedge\mathrm{d}y\\ =\iint_{\Sigma'}(P,Q,R)\cdot[(x_s',y_s',z_s')\times(x_t',y_t',z_t')]\mathrm{d}s\mathrm{d}t\\ =\iint_{\Sigma'}[P(y_s'z_t'-z_s'y_t')+Q(z_s'x_t'-x_s'z_t')+R(x_s'y_t'-y_s'x_t')]\mathrm{d}s\mathrm{d}t ∬Σ​f ​(x,y,z)⋅dσ =∬Σ​Pdy∧dz+Qdz∧dx+Rdx∧dy=∬Σ′​(P,Q,R)⋅[(xs′​,ys′​,zs′​)×(xt′​,yt′​,zt′​)]dsdt=∬Σ′​[P(ys′​zt′​−zs′​yt′​)+Q(zs′​xt′​−xs′​zt′​)+R(xs′​yt′​−ys′​xt′​)]dsdt 特殊地，若积分区域可以表示为 Σ : { x = x y = y z = z ( x , y ) ( ( x , y ) ∈ Σ ′ ) \Sigma:\left\{\begin{array}{l} x=x\\ y=y\\ z=z(x,y) \end{array}\right.\\ ((x,y)\in\Sigma') Σ:⎩⎨⎧​x=xy=yz=z(x,y)​((x,y)∈Σ′) 并规定 ( 1 , 0 , z x ′ ) × ( 0 , 1 , z y ′ ) = ( − z x ′ , − z y ′ , 1 ) (1,0,z_x')\times(0,1,z_y')=(-z_x',-z_y',1) (1,0,zx′​)×(0,1,zy′​)=(−zx′​,−zy′​,1) 的方向为正方向，则有 ∬ Σ f ⃗ ( x , y , z ) ⋅ d σ ⃗ = ∬ Σ P d y ∧ d z + Q d z ∧ d x + R d x ∧ d y = ∬ Σ ′ ( P , Q , R ) ⋅ ( − z x ′ , − z y ′ , 1 ) d x d y = ∬ Σ ′ ( − P z x ′ − Q z y ′ + R ) d x d y \iint_{\Sigma}\vec{f}(x,y,z)\cdot\vec{\mathrm{d}\sigma}=\iint_{\Sigma}P\mathrm{d}y\wedge\mathrm{d}z+Q\mathrm{d}z\wedge\mathrm{d}x+R\mathrm{d}x\wedge\mathrm{d}y\\ =\iint_{\Sigma'}(P,Q,R)\cdot(-z_x',-z_y',1)\mathrm{d}x\mathrm{d}y=\iint_{\Sigma'}(-Pz_x'-Qz_y'+R)\mathrm{d}x\mathrm{d}y ∬Σ​f ​(x,y,z)⋅dσ =∬Σ​Pdy∧dz+Qdz∧dx+Rdx∧dy=∬Σ′​(P,Q,R)⋅(−zx′​,−zy′​,1)dxdy=∬Σ′​(−Pzx′​−Qzy′​+R)dxdy

假定积分区域为 Σ : { r = r ( s , t ) θ = θ ( s , t ) ϕ = ϕ ( s , t ) ( ( s , t ) ∈ Σ ′ ) \Sigma:\left\{\begin{array}{l} r=r(s,t)\\ \theta=\theta(s,t)\\ \phi=\phi(s,t) \end{array}\right.\\ ((s,t)\in\Sigma') Σ:⎩⎨⎧​r=r(s,t)θ=θ(s,t)ϕ=ϕ(s,t)​((s,t)∈Σ′) 并规定 ( r s ′ , θ s ′ , ϕ s ′ ) × ( r t ′ , θ t ′ , ϕ t ′ ) (r_s',\theta_s',\phi_s')\times(r_t',\theta_t',\phi_t') (rs′​,θs′​,ϕs′​)×(rt′​,θt′​,ϕt′​) 的方向为正方向，则有 ∬ Σ f ⃗ ( r , θ , ϕ ) ⋅ d σ ⃗ = ∬ Σ P ( r d θ ) ∧ ( r s i n θ d ϕ ) + Q ( r s i n θ d ϕ ) ∧ d r + R d r ∧ ( r d θ ) = ∬ Σ ′ ( P , Q , R ) ⋅ [ ( r s ′ , r θ s ′ , r s i n θ ϕ s ′ ) × ( r t ′ , r θ t ′ , r s i n θ ϕ t ′ ) ] d s d t = ∬ Σ ′ [ P r 2 s i n θ ( θ s ′ ϕ t ′ − ϕ s ′ θ t ′ ) + Q r s i n θ ( ϕ s ′ r t ′ − r s ′ ϕ t ′ ) + R r ( r s ′ θ t ′ − θ s ′ r t ′ ) ] d s d t \iint_{\Sigma}\vec{f}(r,\theta,\phi)\cdot\vec{\mathrm{d}\sigma}=\iint_{\Sigma}P(r\mathrm{d}\theta)\wedge(r\mathrm{sin}\theta\mathrm{d}\phi)+Q(r\mathrm{sin}\theta\mathrm{d}\phi)\wedge\mathrm{d}r+R\mathrm{d}r\wedge(r\mathrm{d}\theta)\\ =\iint_{\Sigma'}(P,Q,R)\cdot[(r_s',r\theta_s',r\mathrm{sin}\theta\phi_s')\times(r_t',r\theta_t',r\mathrm{sin}\theta\phi_t')]\mathrm{d}s\mathrm{d}t\\ =\iint_{\Sigma'}[Pr^2\mathrm{sin}\theta(\theta_s'\phi_t'-\phi_s'\theta_t')+Qr\mathrm{sin}\theta(\phi_s'r_t'-r_s'\phi_t')+Rr(r_s'\theta_t'-\theta_s'r_t')]\mathrm{d}s\mathrm{d}t ∬Σ​f ​(r,θ,ϕ)⋅dσ =∬Σ​P(rdθ)∧(rsinθdϕ)+Q(rsinθdϕ)∧dr+Rdr∧(rdθ)=∬Σ′​(P,Q,R)⋅[(rs′​,rθs′​,rsinθϕs′​)×(rt′​,rθt′​,rsinθϕt′​)]dsdt=∬Σ′​[Pr2sinθ(θs′​ϕt′​−ϕs′​θt′​)+Qrsinθ(ϕs′​rt′​−rs′​ϕt′​)+Rr(rs′​θt′​−θs′​rt′​)]dsdt

假定积分区域为 Σ : { r = r ( s , t ) ϕ = ϕ ( s , t ) z = z ( s , t ) ( ( s , t ) ∈ Σ ′ ) \Sigma:\left\{\begin{array}{l} r=r(s,t)\\ \phi=\phi(s,t)\\ z=z(s,t)\\ \end{array}\right.\\ ((s,t)\in\Sigma') Σ:⎩⎨⎧​r=r(s,t)ϕ=ϕ(s,t)z=z(s,t)​((s,t)∈Σ′) 并规定 ( r s ′ , ϕ s ′ , z s ′ ) × ( r t ′ , ϕ t ′ , z t ′ ) (r_s',\phi_s',z_s')\times(r_t',\phi_t',z_t') (rs′​,ϕs′​,zs′​)×(rt′​,ϕt′​,zt′​) 的方向为正方向，则有 ∬ Σ f ⃗ ( r , ϕ , z ) ⋅ d σ ⃗ = ∬ Σ P ( r d ϕ ) ∧ d z + Q d z ∧ d r + R d r ∧ ( r d ϕ ) = ∬ Σ ′ ( P , Q , R ) ⋅ [ ( r s ′ , r ϕ s ′ , z s ′ ) × ( r t ′ , r ϕ t ′ , z t ′ ) ] d s d t = ∬ Σ ′ [ P r ( ϕ s ′ z t ′ − z s ′ ϕ t ′ ) + Q ( z s ′ r t ′ − r s ′ z t ′ ) + R r ( r s ′ ϕ t ′ − ϕ s ′ r t ′ ) ] d s d t \iint_{\Sigma}\vec{f}(r,\phi,z)\cdot\vec{\mathrm{d}\sigma}=\iint_{\Sigma}P(r\mathrm{d}\phi)\wedge\mathrm{d}z+Q\mathrm{d}z\wedge\mathrm{d}r+R\mathrm{d}r\wedge(r\mathrm{d}\phi)\\ =\iint_{\Sigma'}(P,Q,R)\cdot[(r_s',r\phi_s',z_s')\times(r_t',r\phi_t',z_t')]\mathrm{d}s\mathrm{d}t\\ =\iint_{\Sigma'}[Pr(\phi_s'z_t'-z_s'\phi_t')+Q(z_s'r_t'-r_s'z_t')+Rr(r_s'\phi_t'-\phi_s'r_t')]\mathrm{d}s\mathrm{d}t ∬Σ​f ​(r,ϕ,z)⋅dσ =∬Σ​P(rdϕ)∧dz+Qdz∧dr+Rdr∧(rdϕ)=∬Σ′​(P,Q,R)⋅[(rs′​,rϕs′​,zs′​)×(rt′​,rϕt′​,zt′​)]dsdt=∬Σ′​[Pr(ϕs′​zt′​−zs′​ϕt′​)+Q(zs′​rt′​−rs′​zt′​)+Rr(rs′​ϕt′​−ϕs′​rt′​)]dsdt