Understanding the Relationship Between Surface & Line Integrals

In summary, Stokes's theorem explains that the surface integral and line integral are equivalent because the integral of ∇×F over a surface has the same units as the integral of F over a curve. This means that the surface integral, which represents area, is equal to the length represented by the line integral. This helps to clarify any confusion about the relationship between the two.
  • #1
hi experts
as far I know the stokes theorem relates surface integral to line integral - but i am confuse how surface integral if represent area gets equal to length as represented by line integral.
 
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  • #2
If the vector field F has units u, then the integral of F over a curve has units u*length. Stokes's theorem says that this is the same as the integral of ∇×F over a surface. ∇×F has units of u/length (since each of the partial derivatives does), so the integral of that over a surface has units u/length * length^2 = u*length - just as the integral of F over a curve does. Does that help?
 
  • #3
great - thanks alot
 

What is the difference between surface and line integrals?

Surface integrals are used to calculate the total flux or flow through a two-dimensional surface, while line integrals are used to calculate the total change along a one-dimensional curve or path.

How are surface and line integrals related?

Surface integrals can be thought of as the sum of multiple line integrals, as the surface is made up of infinitely many curves. Additionally, line integrals can be used to find the boundary of a surface integral.

What are some real-world applications of surface and line integrals?

Surface integrals are used in fluid mechanics to calculate fluid flow through a surface, in electromagnetism to calculate electric and magnetic fields, and in computer graphics to calculate light and color on a surface. Line integrals are used in physics to calculate work and energy along a path, in engineering to calculate force and torque along a path, and in economics to calculate production and consumption along a production possibility curve.

What is the difference between a closed and open surface in relation to surface integrals?

A closed surface is one that completely encloses a volume, while an open surface does not enclose a volume. Closed surfaces are used in surface integrals to calculate the total flux through a volume, while open surfaces are used to calculate the flux through a surface.

How can surface and line integrals be calculated?

Surface integrals can be calculated using double integrals, where the surface is represented by a function of two variables. Line integrals can be calculated using single integrals, where the curve is represented by a function of one variable. In both cases, the integral is solved by breaking the surface or curve into smaller, more manageable pieces and summing them together.

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