Divergence Theorem Calculator
Verify the divergence theorem (Gauss's theorem) by computing flux and volume integrals.
Vector Field F(x, y, z)
Region
Total volume evaluations: ~27,000
Divergence Theorem
oiint_S F dot dS = iiint_V (div F) dV
Theorem Verification
Verified
Difference: 0.003492
oiintFlux (Surface)
12.572114
iiintVolume Integral
12.568622
VVolume
4.189541
ASurface Area
12.572114
Calculation Details
Left side (outward flux):
oiint_S F dot n dS = 12.572114
Right side (divergence):
iiint_V (nabla dot F) dV = 12.568622
Physical Meaning
- Net outward flux = total source strength inside
- Positive div: sources (fluid expanding)
- Negative div: sinks (fluid contracting)
- Zero div: incompressible flow
About the Divergence Theorem
Statement
The divergence theorem (Gauss's theorem) relates the flux of a vector field through a closed surface to the volume integral of the divergence over the enclosed region.
Applications
- Gauss's law in electrostatics
- Conservation laws in physics
- Fluid flow analysis
- Heat transfer calculations