Flux/ Divergence Theorem interpretation

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SUMMARY

The discussion centers on the interpretation of flux and the divergence theorem in multivariable calculus. Flux, represented as ∫∫ F • N dS, quantifies fluid flow through a surface S per unit time, while the divergence theorem, expressed as ∫∫∫ divF dV, measures fluid flow through a solid region. The confusion arises from the terminology, where "across" is clarified to mean "through," aligning the two concepts. This semantic distinction is crucial for understanding the relationship between surface integrals and volume integrals in vector analysis.

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leehufford
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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
 
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Hi Lee! Thankfully this just seems to be a problem with the usage of English phrases. "Across" here means "through" and not "over".
 
Its like "running across the finish line".
 

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