Surface Integral Calculator
Calculate surface integrals and flux of vector fields over parametric surfaces.
Vector Field F(x, y, z)
Parametric Surface r(u, v)
Parameter Ranges
Surface Integral (Flux)
1.000000
integral_S F dot dS
ASurface Area
1.861511
PhiFlux
1.000000
Surface Integral Formula
integral_S F dot dS = integral integral_D F(r(u,v)) dot (r_u x r_v) du dv
Interpretation
- Flux: Net flow of vector field through surface
- Positive flux: Field flows outward (same direction as normal)
- Negative flux: Field flows inward (opposite to normal)
- Zero flux: Equal inflow and outflow
Applications
- Electric flux (Gauss's law)
- Magnetic flux through surfaces
- Fluid flow rates
- Heat transfer through boundaries
About Surface Integrals
Definition
A surface integral extends the concept of line integrals to two dimensions. The flux integral measures the total flow of a vector field through a surface.
Related Theorems
- Stokes' theorem: relates to line integrals
- Divergence theorem: relates to volume integrals
- Green's theorem: 2D special case