Kinematics of surface integrals question

[skybobster]

This is a continuum mechanics/fluid dynamics question concerning the time rate of change of a surface integral of a vector field, where the surface is flowing along in a velocity field (like in a fluid). (Gauss's law is for fixed surfaces.) This integral goes by various names in different applications, like "Faraday's law for moving media" in electromagnetism, or "Zorawski's criterion" in fluid dynamics (when the integral vanishes). (Truesdell, "Kinematics of Vorticity" page 55). We have two vector fields, one is the velocity vector field, and the other is some other specified vector field, like the magnetic field in E-M. Most of the derivations are based on a surface patch flowing along in the fluid (or more generally a velocity field), this patch surrounded by a closed curve. When the integral vanishes, it means that the integral of the given vector field dotted into the moving surface element of the (velocity) vector-tube (the tube swept out by the flowing surface patch) for the given vector field remains the same during the motion, and we end up with an elegant vector equation (called sometimes Zorawski's criterion). My question is this: Is this also true for closed moving surfaces (instead of a surface patch) like for example a deformed sphere flowing along? Nowhere in the literature can I find this important case of a closed moving surface.

 

[fresh_42]

Why should it be treated differently? You can still consider it as a differential manifold, closed or not.
Maybe you could use some keywords from https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/ for further searches.