Flux/ Divergence Theorem interpretation

In summary, the difference between flux and the divergence theorem lies in the perspective of fluid flow - flux measures flow across a surface while the divergence theorem measures flow through a solid. This distinction can be confusing due to the use of the phrase "across", which actually means "through".
  • #1
leehufford
98
1
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 [itex]\bullet[/itex] 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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  • #2
Hi Lee! Thankfully this just seems to be a problem with the usage of English phrases. "Across" here means "through" and not "over".
 
  • #3
Its like "running across the finish line".
 

Related to Flux/ Divergence Theorem interpretation

1. What is the Flux/Divergence Theorem?

The Flux/Divergence Theorem, also known as Gauss's Theorem, is a fundamental concept in vector calculus that describes the relationship between a vector field and its sources and sinks. It states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the field within the surface.

2. How is the Flux/Divergence Theorem used in physics?

In physics, the Flux/Divergence Theorem is used to calculate the flow of a vector field through a given surface. It is particularly useful in electromagnetism, fluid dynamics, and other areas where vector fields play a significant role.

3. Can you explain the intuition behind the Flux/Divergence Theorem?

The Flux/Divergence Theorem can be thought of as a generalization of the fundamental theorem of calculus. It states that the total change in a vector field over a given region is equal to the sum of the flux through the boundary of that region and the sources and sinks within the region. In other words, it connects the local behavior of a vector field to its global properties.

4. What is the relationship between the Flux/Divergence Theorem and the Divergence Theorem?

The Flux/Divergence Theorem is essentially a higher-dimensional version of the Divergence Theorem. While the Divergence Theorem applies to three-dimensional vector fields, the Flux/Divergence Theorem can be extended to any number of dimensions. Both theorems describe the relationship between a vector field and its divergence, but the Flux/Divergence Theorem includes the concept of flux through a surface.

5. How is the Flux/Divergence Theorem applied in real-world situations?

The Flux/Divergence Theorem has many practical applications, such as calculating the flow of air or water through a given surface, predicting the behavior of electric and magnetic fields, and understanding the dynamics of fluids in pipes or channels. It is also used in image processing and computer graphics to analyze vector fields and their behavior.

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