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Analytic Geometry2D

Cylindrical and Spherical Coordinates - Theory & Concepts - Analytic Geometry Spherical Volume Element

Advanced 3D coordinate systems and their relationship with rectangular coordinates.

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Spherical Volume Element (dV) Explorer

Position Variables
Radius (ρ)3.00
Zenith Angle (φ)45°
Azimuth Angle (θ)60°
Differentials (Size)
Thickness (dρ)0.80
Delta Zenith (dφ)20°
Delta Azimuth (dθ)25°
Approx dV:0.7754
Exact Volume V:1.1533
Difference:32.76%
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Spherical Volume Integration
dV=ρ2sin⁡ϕ dρ dϕ dθdV = \rho^2 \sin\phi \, d\rho \, d\phi \, d\theta
Substitution & Calculations
Approx dV≈(3.00)2sin⁡(45∘)⋅(0.80)⋅(0.3491 rad)⋅(0.4363 rad)=0.77543\text{Approx dV} \approx (3.00)^2 \sin(45^\circ) \cdot (0.80) \cdot (0.3491\text{ rad}) \cdot (0.4363\text{ rad}) = 0.77543Exact Volume V=∫θθ+dθ ⁣∫ϕϕ+dϕ ⁣∫ρρ+dρ ⁣r2sin⁡ϕ dr dϕ dθ=(30.80)3−(3.00)33[cos⁡(45∘)−cos⁡(65∘)](0.4363)=1.15326\text{Exact Volume } V = \int_{\theta}^{\theta+d\theta}\!\int_{\phi}^{\phi+d\phi}\!\int_{\rho}^{\rho+d\rho}\! r^2\sin\phi\, dr\,d\phi\,d\theta = \frac{(30.80)^3 - (3.00)^3}{3} \Big[ \cos(45^\circ) - \cos(65^\circ) \Big] (0.4363) = 1.15326

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