[NP] Basic Background of Fluid Dynamics

To understand Navier-Stokes equation, we need plenty of background knowledge from physics. Here are some easiest ones. Concepts:

• Flux: the amount of stuff travel through a surface.
• Divergence: measure how powerful the flux is in a field. Topological definition is the limit of surface integral around the given point devided by contained volume, with the volume shrinking to 0 towards that point. With a coordinate, divergence is the dot product of gradient operator with the vector field.
• Circulation: the amount of stuff wind around a line.
• Curl: measure how powerful the circulation is in a field. Topological definition is the limit of line integral around the circle devided by surrounding area, where the area approching to 0 with the circle shrinking to its central line. With a coordinate, curl is the cross product of gradient operator with the vector field.

Theorems:

• Stokes theorem: The closed line integral of a vector field equals to the surface integral of the curl of this vector field on any surface surrounded by this closed line.
• Green theorem: 2 dimensional version of Stokes theorem. Just set the 3rd component of the vector field to be 0.
• Divergence theorem: The triple integral of the divergence of a vector field on a region equals to the surface integral of the vector field on the region's boundary.
• Their relationship can be found here.

* Those are only basic MATH 2E knowledge, i.e. multi-variable calculus.