Flux is integral of divergence over surface¶
The flux of a vector field exiting a closed flat curve can be re-written as a surface integral of the divergence of the given field over the surface whose boundary is the given curve.
Notes:
This is also known as the Green’s theorem.
Conditions:
The vector field is continuously differentiable everywhere within the region of the surface.
The curve is closed and flat.
The curve and the surface are oriented according to the right-hand rule.
- Symbol:
H- Latex:
\(H\)
- Dimension:
any_dimension
- field¶
Any vector field, i.e. a vector-valued function that depends on the position vector.
- Symbol:
F- Latex:
\({\vec F}\)
- Dimension:
any_dimension
- surface¶
Surface along which the integral is evaluated.
- Symbol:
S- Latex:
\(S\)
- initial_first_parameter¶
Initial value of the first surface parameter.
- Symbol:
u_1- Latex:
\(u_{1}\)
- Dimension:
dimensionless
- final_first_parameter¶
Final value of the first surface parameter.
- Symbol:
u_2- Latex:
\(u_{2}\)
- Dimension:
dimensionless
- initial_second_parameter¶
Initial value of the second surface parameter.
- Symbol:
v_1- Latex:
\(v_{1}\)
- Dimension:
dimensionless
- final_second_parameter¶
Final value of the second surface parameter.
- Symbol:
v_2- Latex:
\(v_{2}\)
- Dimension:
dimensionless
- law¶
H = SurfaceIntegral(div(F) * norm(dS), S, ((u_1, u_2), (v_1, v_2)))Latex:
\[H = \iint \limits_S \text{div} \vec F \, dS\]