Discover the power of the Gauss-Ostrogradsky Divergence Theorem, also known as Green's Theorem in 3D. This calculator helps you understand its application in vector calculus, bridging surface integrals with volume integrals. Explore fundamental concepts, verify proofs, and work through examples to solidify your grasp of this crucial mathematical theorem.
Formula:
The Gauss-Ostrogradsky Divergence Theorem states that the flux of a vector field F through a closed surface S is equal to the triple integral of the divergence of F over the volume V enclosed by S.
Mathematically: ∮S F ⋅ dS = ∫∫∫V (∇ ⋅ F) dV
This calculator applies the theorem to the specific example of a vector field F = <x, y, z> over a sphere of radius R centered at the origin.
In this case:
- Divergence (∇ ⋅ F) = ∂x/∂x + ∂y/∂y + ∂z/∂z = 3
- Volume Integral: ∫∫∫V 3 dV = 3 × (Volume of Sphere) = 3 × (4/3)πR³ = 4πR³
- Surface Integral: ∮S F ⋅ dS = ∮S (xi + yj + zk) ⋅ (x/R i + y/R j + z/R k) dS = ∮S (x² + y² + z²)/R dS = ∮S R²⁄R dS = ∮S R dS = R × (Surface Area of Sphere) = R × 4πR² = 4πR³