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Is it correct to interpret the surface integral of a vector function $\mathbf{v}$ over four sides of a cube as the rate of flow of fluid (in mass per unit time) that would flow out of the cube when those three sides are opened, given that the cube has an "infinite" amount of fluid (so it won' t run out), and that $\mathbf{v}$ gives the rate of flow of fluid in mass per unit time per unit area?

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What you are looking for is called flux, specifically this definition.

Answer is, yes. The $\mathbf{v}$, usually denoted $\mathbf{j}$, is named the current density in electrostatics and simply flux (the other definition of it, anyways) in other fields, and you can define it for mass, electrical charge, heat, number of particles etc. It's defined as amount of matter that will pass through a infinitesimal surface with normal vector $\hat{\mathbf{j}}$ per infinitesimal time divided by the surface and time (notice this definition matches yours).

Furthermore, as per the continuity equation, you can connect this integral to the actual change in quantity over a specific volume as well.

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  • $\begingroup$ Thanks! This was a huge boost to my confidence! I'll accept your answer. $\endgroup$ – Christoffer Corfield Aakre Jul 7 at 20:06

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