Animating Normal (Maps)

There seems to be an on-going confusion on how to animate normal maps. The best answer to this is: you don't :-).
The obvious problem is to stream in two normal maps and then to modulate two normals. If you are on a console platform you just don't want to do this. So what would be a good way to animate a normal? You modulate height fields. Where both height fields have peaks, the result should also have a peak. Where one of the height fields is zero, the result should be also be zero, independent of the other height field.

Usually, a normal map is formed by computing a relief of a height field (bump map) over a flat surface. If is the bump map’s height at the texture coordinates, the standard definition of the normal map is

The height fields are multiplied to form a combined height field like this

To determine the normal vector of this height field according to the first Equation, one needs the partial derivatives of this functions. This is a simple application of the product rule:

And similarly for the partial derivative with respect to v. Thus:

BTW: to recover the height field’s partial derivatives from the normal map we can use: