Circulation is integral of curl over surface¶
The circulation of a vector field, defined as such, can be re-written as a double integral of the flux of that field over any surface whose boundary is the curve along which the circulation is calculated.
Notes:
This law is also known as the Stokes’ theorem.
Conditions:
The surface is smooth.
The orientation of the surface and the orientation of the curve are related by the right-hand rule.
- Symbol:
G- Latex:
\(G\)
- 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¶
G = SurfaceIntegral(dot(curl(F), dS), S, ((u_1, u_2), (v_1, v_2)))Latex:
\[G = \iint \limits_S \left( \nabla \times \vec F \right) \cdot d \vec S\]