We get to that point, c++, this chapter mainly introduce parts of c++, and understand them with many code example.

We recall the continuity and Euler momentum equations in Eulerian form, derive the material derivative \(\mathrm{d} \mathbf{v} / \mathrm{d} t=\partial_t \mathbf{v}+(\mathbf{v} \cdot \nabla) \mathbf{v}\), and explain how Eulerian field descriptions relate to Lagrangian particlebased descriptions via \(\rho(\mathbf{x}(t), t)\).

This note primarily delves into incompressible flow momentum equation and Surface of water waves and boundary condition and Variational Formulation.

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We collect working formulas for dyadic (outer) products, the double-dot contraction of tensors, and tensor–vector cross products expressed with the Levi-Civita symbol, and we recall the construction of surface elements in spherical coordinates. We also note some standard Levi-Civita identities and briefly relate tensor gradients and variational increments to familiar differential operators.