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i-m-Calculated-gauess-law-in-Gravity-Field-S-g-dS-g-x-y-z-Gmx-x-2-y-2-z-2-e-1-Gmy-x-2-y-2-z-2-e-2-Gmz-x-2-y-2-z-2-e-3-S-is-Clo

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Question Number 199157 by MathedUp last updated on 30/Oct/23
i′m Calculated gauess law in Gravity Field ∫∫_( S) g^� ∙dS^� g^� (x,y,z)=−((Gmx)/( (√(x^2 +y^2 +z^2 ))))e_1 ^� −((Gmy)/( (√(x^2 +y^2 +z^2 ))))e_2 ^� −((Gmz)/( (√(x^2 +y^2 +z^2 ))))e_3 ^� S is Closed Circle , x^2 +y^2 +z^2 =r^2 x=rsin(θ)cos(ρ) y=rsin(θ)sin(ρ) z=rcos(θ) r^� (θ,ρ)=rsin(θ)cos(ρ)e_1 ^� +rsin(θ)sin(ρ)e_2 ^� +rcos(θ)e_3 ^� g^� (θ,ρ)=sin(θ)cos(ρ)e_1 ^� +sin(θ)sin(ρ)e_2 ^� +cos(θ)e_3 ^� (∂r^� /∂θ)×(∂r^� /∂ρ)= determinant ((( e_1 ^� ),( e_2 ^� ),( e_3 ^� )),(( rcos(θ)cos(ρ)),(rcos(θ)sin(ρ)),(−rsin(θ))),((−rsin(θ)sin(ρ)),(rsin(θ)cos(ρ)),( 0))) ∴r^2 sin^2 (θ)cos(ρ)e_1 ^� +r^2 sin^2 (θ)sin(ρ)e_2 ^� +r^2 sin(θ)cos(θ)e_3 ^� −(r^2 sin^3 (θ)cos^2 (ρ)+r^2 sin^3 (θ)sin^2 (ρ)+r^2 sin(θ)cos^2 (θ))dθdρ −(r^2 sin^3 (θ)+r^2 sin(θ)cos^2 (θ))dθdρ=−r^2 sin(θ)dθdρ −∫∫_( S) r^2 sin(θ)dθdρ=−∫_0 ^( 2π) ∫_0 ^( π) r^2 sin(θ)dθdρ ∴ ∫∫_( S) g^� ∙dS^� =−4πGmr^2 r=1 ∫∫_( S) g^� ∙dS^� =−4πGm ∫∫_( D) ▽^� ∙g^� dA=−4πG∫∫_( D) 𝛒 dA ▽^� ∙g^� =−4π𝛒G
$$\mathrm{i}'\mathrm{m}\:\mathrm{Calculated}\:\:\mathrm{gauess}\:\mathrm{law}\:\mathrm{in}\:\mathrm{Gravity}\:\mathrm{Field} \\ $$$$ \\ $$$$\int\int_{\:\boldsymbol{{S}}} \:\hat {\boldsymbol{\mathrm{g}}}\centerdot\mathrm{d}\hat {\boldsymbol{\mathrm{S}}} \\ $$$$\hat {\boldsymbol{\mathrm{g}}}\left({x},{y},{z}\right)=−\frac{{Gmx}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} −\frac{{Gmy}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} −\frac{{Gmz}}{\:\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} }}\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\boldsymbol{{S}}\:\mathrm{is}\:\mathrm{Closed}\:\mathrm{Circle}\:,\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} +{z}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$${x}={r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right) \\ $$$${y}={r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right) \\ $$$${z}={r}\mathrm{cos}\left(\theta\right) \\ $$$$\hat {\boldsymbol{\mathrm{r}}}\left(\theta,\rho\right)={r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +{r}\mathrm{cos}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\hat {\boldsymbol{\mathrm{g}}}\left(\theta,\rho\right)=\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +\mathrm{cos}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$\frac{\partial\hat {\boldsymbol{\mathrm{r}}}}{\partial\theta}×\frac{\partial\hat {\boldsymbol{\mathrm{r}}}}{\partial\rho}=\:\begin{vmatrix}{\:\:\:\:\:\:\:\:\:\:\:\:\:\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} }&{\:\:\:\:\:\:\:\:\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} }&{\:\:\:\:\:\:\:\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} }\\{\:\:\:\:{r}\mathrm{cos}\left(\theta\right)\mathrm{cos}\left(\rho\right)}&{{r}\mathrm{cos}\left(\theta\right)\mathrm{sin}\left(\rho\right)}&{−{r}\mathrm{sin}\left(\theta\right)}\\{−{r}\mathrm{sin}\left(\theta\right)\mathrm{sin}\left(\rho\right)}&{{r}\mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\rho\right)}&{\:\:\:\:\:\:\:\mathrm{0}}\end{vmatrix} \\ $$$$\therefore{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)\mathrm{cos}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{1}} +{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \left(\theta\right)\mathrm{sin}\left(\rho\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{2}} +{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}\left(\theta\right)\hat {\boldsymbol{\mathrm{e}}}_{\mathrm{3}} \\ $$$$−\left({r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{3}} \left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\rho\right)+{r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{3}} \left(\theta\right)\mathrm{sin}^{\mathrm{2}} \left(\rho\right)+{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\theta\right)\right)\mathrm{d}\theta\mathrm{d}\rho \\ $$$$−\left({r}^{\mathrm{2}} \mathrm{sin}^{\mathrm{3}} \left(\theta\right)+{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{cos}^{\mathrm{2}} \left(\theta\right)\right)\mathrm{d}\theta\mathrm{d}\rho=−{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{d}\theta\mathrm{d}\rho \\ $$$$−\int\int_{\:\boldsymbol{\mathcal{S}}} \:\:{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{d}\theta\mathrm{d}\rho=−\int_{\mathrm{0}} ^{\:\mathrm{2}\pi} \int_{\mathrm{0}} ^{\:\pi} \:{r}^{\mathrm{2}} \mathrm{sin}\left(\theta\right)\mathrm{d}\theta\mathrm{d}\rho \\ $$$$\therefore\:\int\int_{\:\boldsymbol{\mathcal{S}}} \:\hat {\boldsymbol{\mathrm{g}}}\centerdot\mathrm{d}\hat {\boldsymbol{\mathrm{S}}}=−\mathrm{4}\pi{Gmr}^{\mathrm{2}} \:\: \\ $$$${r}=\mathrm{1}\:\:\int\int_{\:\boldsymbol{\mathcal{S}}} \:\hat {\boldsymbol{\mathrm{g}}}\centerdot\mathrm{d}\hat {\boldsymbol{\mathrm{S}}}=−\mathrm{4}\pi{Gm}\:\:\:\:\: \\ $$$$\int\int_{\:\:\boldsymbol{\mathcal{D}}} \:\hat {\bigtriangledown}\centerdot\hat {\boldsymbol{\mathrm{g}}}\:\mathrm{dA}=−\mathrm{4}\pi{G}\int\int_{\:\boldsymbol{\mathcal{D}}} \:\boldsymbol{\rho}\:\mathrm{dA} \\ $$$$\hat {\bigtriangledown}\centerdot\hat {\boldsymbol{\mathrm{g}}}=−\mathrm{4}\pi\boldsymbol{\rho}{G} \\ $$

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