Flux/ Divergence Theorem interpretation

[leehufford]

Hello,

I am approaching the end of my multivariable/ vector analysis "Calc III" class and have a question about flux.

My book states that flux, ∫∫ F \bullet N dS measures the fluid flow "across" a surface S per unit time.

Now, the divergence theorem ∫∫∫ divF dV measures the "same thing" but fluid flow THROUGH a solid. How can fluid flow over a surface and through a solid be equated for the same regoin? Is it actually fluid flow THROUGH a surface and not OVER it? That would make more sense.

Hopefully some people who are further along in their math journey can explain this for me. Thanks in advance,

Lee

 

[WannabeNewton]

Hi Lee! Thankfully this just seems to be a problem with the usage of English phrases. "Across" here means "through" and not "over".

 

[HallsofIvy]

Its like "running across the finish line".