This lesson covers the development of important mathematical tools used in aerodynamics. It delves into the concepts of circulation, velocity potential, and conservation equations of flow field. The lesson further discusses line, surface, and volume integrals and their relationships. It also explains the connection between circulation and vorticity, and the concept of velocity potential in irrotational flows. For instance, the lesson illustrates how a three-dimensional velocity field can be represented by a single variable, Phi, through directional derivatives.

- Circulation is a kinematic property that depends on the velocity field and the choice of the curve. It is linked with aerodynamic lift and is connected with vorticity, which has to do with the rotationality of the flow field.
- A three-dimensional velocity field can be represented purely by one variable, Phi, if it satisfies the irrotationality condition.
- Stokes theorem connects line integral with surface integral. It states that the line integral of a vector field around a closed curve is equal to the surface integral of the curl of the vector field over the surface enclosed by the curve.
- Gradient theorem connects surface integral and volume integral. It states that the surface integral of a scalar field over a closed surface is equal to the volume integral of the gradient of the scalar field over the volume enclosed by the surface.

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