This lesson covers the conservation laws in hydrodynamics, focusing on the quadratic quantities that are crucial for describing fluid flows. The lesson explains the concepts of kinetic energy density, kinetic helicity, and enstrophy, and how they are constructed using velocity fields. It also discusses the evolution equation for kinetic energy density and how it can be derived from the Navier-Stokes equation. The lesson further explores the integral form of the conservation laws and the conditions under which certain quantities are conserved. For instance, kinetic energy is conserved for both 2D and 3D, kinetic helicity is conserved in 3D, and enstrophy is conserved only in 2D. The lesson concludes with a discussion on the concept of circulation and its conservation.

- Kinetic energy density, kinetic helicity, and enstrophy are quadratic quantities that are important in describing fluid flows.
- The evolution equation for kinetic energy density can be derived from the Navier-Stokes equation.
- The conservation laws can be expressed in integral form.
- Kinetic energy is conserved for both 2D and 3D, kinetic helicity is conserved in 3D, and enstrophy is conserved only in 2D.
- The concept of circulation is also important in hydrodynamics, and its conservation is discussed.

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