What does the new integral in surface integral theory represent?

In summary, the conversation discusses the concept of a surface integral, which involves assigning a value to each part of a surface based on a new function f(x,y,z). The speaker presents three ideas for what this integral could represent, including a volume between two surfaces, the surface area of the new surface f(x,y,z), and something in 4D. They also mention the possibility of using a parametric surface to map directly to the new surface. Finally, they mention the concept of a variable mass density and how it could be used in a surface integral to calculate the mass of the surface.
  • #1
andyb177
10
0
Watching video http://www.khanacademy.org/video/introduction-to-the-surface-integral?playlist=Calculus
at 20.10 the guy introduces the concept of what it means for each part of the surface to have a value of a new function f(x,y,z). Could some one explain what this new integral would represent?

My Three ideas are..
A volume between the two surfaces,
Just the surface area of the new surface f(x,y,z) which takes value from the param. surface in order to 'create it self'?
If the second one (which I think is more likely) why not create a param. to map straight to this surface?
Or is it something in 4d?

Thanks alot.
 
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  • #2
andyb177 said:
Watching video http://www.khanacademy.org/video/introduction-to-the-surface-integral?playlist=Calculus
at 20.10 the guy introduces the concept of what it means for each part of the surface to have a value of a new function f(x,y,z). Could some one explain what this new integral would represent?

My Three ideas are..
A volume between the two surfaces,
Just the surface area of the new surface f(x,y,z) which takes value from the param. surface in order to 'create it self'?
If the second one (which I think is more likely) why not create a param. to map straight to this surface?
Or is it something in 4d?

Thanks alot.

I didn't want to bother watching the video. But an example of what I think you are describing would be a surface with a variable area mass density δ(x,y,z) given in units like, for example, kg/m2. Then

[tex]\iint_S \delta(x,y,z)\, dS[/tex]

would represent the mass of the surface in kg.
 
  • #3
Bang on, cheers
 

FAQ: What does the new integral in surface integral theory represent?

What is surface integral theory?

Surface integral theory is a mathematical concept used in various fields of science, such as physics and engineering. It involves calculating the flux or flow of a vector field through a surface, and is typically used to solve problems involving fluid flow or electromagnetism.

How is surface integral theory different from line integral theory?

The main difference between surface integral theory and line integral theory is the dimensionality of the objects being integrated over. Surface integrals are performed over two-dimensional surfaces, while line integrals are performed over one-dimensional curves.

What are some real-world applications of surface integral theory?

Surface integral theory is used in a variety of fields, including fluid dynamics, electromagnetism, and heat transfer. Some specific examples of applications include calculating the flow of a fluid through a pipe, determining the electric field around a charged object, and analyzing the transfer of heat through a material.

How is surface integral theory related to the fundamental theorem of calculus?

The fundamental theorem of calculus states that the integral of a function can be calculated as the difference between its values at the endpoints. Surface integral theory extends this concept to calculate the flux or flow of a vector field through a surface by integrating over infinitesimal elements of the surface.

Are there any limitations to surface integral theory?

Like any mathematical theory, surface integral theory has its limitations. It is most useful for solving problems involving steady-state conditions, and may not be as accurate for transient or dynamic situations. Additionally, it may be difficult to apply in cases where the surface is irregular or the vector field is not well-behaved.

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