Surface Integral: Right Side = Left Side?

In summary, when dealing with the equation \boxed{\int_{S} \nabla \times \vec{B}.d\vec{a}=\oint \vec{B}.d\vec{l}},it doesn't matter which surface is chosen as long as the boundary remains the same. This allows for flexibility and ease in calculations depending on the symmetry of the chosen surface. There is a proof for this theorem, but it may not be readily available.
  • #1
pardesi
339
0
May be this should have been in math section but since this came out while studying Electrodynamics i put it here
we have
[tex]\boxed{\int_{S} \nabla \times \vec{B}.d\vec{a}=\oint \vec{B}.d\vec{l}}[/tex]

Q.well there are many areas with the same boundary which one to choose from?

well if we know the area the boundary is fixed but not vice-versa does only the right side equal left but nor always the left side equals right.
Can someone explain
 
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  • #2
It doesn't matter which surface you take, that's the beauty of it. For example, if the curve on the right hand side is a circle, depending on the symmetry it may be easiest to take either a flat disk, or a half-sphere, or anything else.

There's probably a nice proof for it too, but I wouldn't be able to give you that by heart.
 
  • #3
wow
tat's gr8
 
  • #4
CompuChip said:
It doesn't matter which surface you take, that's the beauty of it. For example, if the curve on the right hand side is a circle, depending on the symmetry it may be easiest to take either a flat disk, or a half-sphere, or anything else.

There's probably a nice proof for it too, but I wouldn't be able to give you that by heart.

Stokes's Theorem.
 

Related to Surface Integral: Right Side = Left Side?

1. What is the definition of a surface integral?

A surface integral is a mathematical concept used in calculus to calculate the total value of a function over a two-dimensional surface. It can be seen as the equivalent of a double integral in three-dimensional space.

2. What is the difference between a surface integral and a line integral?

A surface integral is used to find the total value of a function over a two-dimensional surface, while a line integral is used to find the total value of a function along a one-dimensional curve. In other words, a surface integral integrates over a surface, while a line integral integrates along a curve.

3. How do you calculate a surface integral?

To calculate a surface integral, you first need to parameterize the surface into a two-dimensional region. Then, you integrate the function over this region, using a double integral. This can be done using various methods, such as the parametric surface area formula or Green's theorem.

4. What is the significance of the right side being equal to the left side in a surface integral?

When the right side of a surface integral is equal to the left side, it means that the function is being integrated correctly over the given surface. This is important because it ensures that the calculated value is accurate and reflects the true value of the function over the surface.

5. What are some real-life applications of surface integrals?

Surface integrals have many practical applications, such as calculating the flux of a vector field through a surface, finding the surface area of a curved object, and determining the mass or center of mass of a three-dimensional object. They are also used in fields such as physics, engineering, and computer graphics.

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