What is the Coarea Formula and How Can It Be Proven?

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In summary, the conversation discusses the attempt to prove the coarea formula with the help of an appropriate change of variable. The person is seeking hints or a link to a relevant source, but has not found anything outside of the Wikipedia page on coarea. They suggest that section 4.2 of a certain paper may be relevant. Later, they mention that they have figured it out using integration over manifolds and Fubini's theorem.
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Matthollyw00d
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I'm trying to prove the coarea formula (not in the most general case) and it looks like it shouldn't be too difficult after an appropriate change of variable, however I'm not seeing that appropriate change of variable. Any hints? A link would work fine too, however outside the wikipedia page on coarea, I've yet to come across any site with anything relevant at all. For such an important formula, there sure is nothing on it.
 
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I figured it out with a little integration over manifolds and fubini's theorem. Thanks anyway though.
 

FAQ: What is the Coarea Formula and How Can It Be Proven?

What is the coarea formula?

The coarea formula is a mathematical theorem that relates the area of a surface to the volume of a higher-dimensional region. It is commonly used in calculus and differential geometry to solve problems involving integration over surfaces.

How is the coarea formula derived?

The coarea formula is derived using the concept of differential forms, specifically the exterior derivative. It involves breaking down a higher-dimensional region into infinitesimal slices and integrating over those slices to find the total volume.

What is the significance of the coarea formula?

The coarea formula is significant because it allows for the simplification of complex integration problems involving surfaces. It also has applications in physics, such as calculating the flux of a vector field through a surface.

What are the limitations of the coarea formula?

The coarea formula is limited in its applicability to certain types of surfaces and regions. It may also be difficult to use in cases where the higher-dimensional region is not well-defined or is irregular in shape.

How is the coarea formula used in real-world applications?

The coarea formula has many real-world applications, such as in computer graphics for rendering 3D objects, in fluid dynamics for calculating flow rates over surfaces, and in economics for measuring the distribution of wealth over a geographic region.

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