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.

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.

We review the Levi-Civita symbol in arbitrary dimension, its role in defining the cross product and wedge product, introduce the Hodge star on differential forms, and clarify the relation between metric components, coordinate bases, and orthonormal frames (with spherical coordinates as a concrete example).

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