Matematika/Gauso formulė: Skirtumas tarp puslapio versijų
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32 eilutė:
:Patikrinsime apskaičiuodami <math>\iint_S x^3 \mathbf{d}y \mathbf{d}z, \; \iint_S y^3 \mathbf{d}z \mathbf{d}x</math> ir <math>\iint_S x^3 \mathbf{d}x \mathbf{d}y</math> sumą.
:<math>x^2=R^2-y^2-z^2,</math>
:<math>x=\sqrt{R^2-y^2-z^2}
:<math>\frac{\partial x}{\partial y}=\frac{\partial(\sqrt{R^2-y^2-z^2})}{\partial y}=\frac{-2 y}{2\sqrt{R^2-y^2-z^2}}
:<math>\frac{\partial x}{\partial z}=\frac{\partial(\sqrt{R^2-y^2-z^2})}{\partial z}=\frac{-2 z}{2\sqrt{R^2-y^2-z^2}}
:<math>V_x=\iint_S x^3 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-y^2-z^2} \right)^3 \mathbf{d}y \mathbf{d}z=\iint_S \left( \sqrt{R^2-\rho^2} \right)^3 \mathbf{d}\rho \mathbf{d}\phi=</math>
:<math>=\int_0^{2\pi} \left( \int_0^R \sqrt{(R^2-\rho^2)^3} \mathbf{d}\rho \right) \mathbf{d}\phi .</math>
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