How to Apply Stoke's Theorem on a Hemispherical Surface?

In summary, the conversation discusses how to approach a question involving ∫F.dr and the use of surface projections. The question asks how to determine the surface for C, and the author suggests using the xy-plane projection. The solution provided in the 4th photo also uses the xy-plane projection, indicating that the concept is correct.
  • #1
fonseh
529
2

Homework Statement


In the first and second photo , it's stated earlier that the C is the boundary of surface on xy plane , but in the question in the 3rd picture , it's not stated that the C is on which surface , so , how to do this question ?
For ∫F.dr , i am not sure how to get r , coz i am not sure i should to use which surface as 2d projection .
In the first photo , it's stated earlier that C is on xy-plane , so xy - plane projection is used.
In the solution provided in the 4th photo , the author use xy plane projection , is the concept correct ?

Homework Equations

The Attempt at a Solution

 

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  • #2
fonseh said:
in the question in the 3rd picture , it's not stated that the C is on which surface
Not an area I know, but it looks to me that you can define C as the boundary in xy plane and that allows you to apply Stokes' theorem.
 

1. What is Stoke's theorem of hemisphere?

Stoke's theorem of hemisphere is a mathematical theorem that states the relationship between a closed surface integral and a surface integral over the boundary of the surface. It is named after mathematician George Gabriel Stokes.

2. How is Stoke's theorem of hemisphere used in physics?

Stoke's theorem of hemisphere is commonly used in physics to calculate the flow of a vector field through a surface. It helps to simplify complex integrals and makes it easier to solve problems involving fluid dynamics, electromagnetism, and other physical phenomena.

3. What are the assumptions of Stoke's theorem of hemisphere?

The main assumptions of Stoke's theorem of hemisphere are that the surface is smooth and orientable, and that the vector field is continuous and differentiable. Additionally, the surface must be closed and have a boundary that is a smooth curve.

4. Can Stoke's theorem of hemisphere be applied to any shape?

No, Stoke's theorem of hemisphere can only be applied to surfaces that are smooth and orientable. It also requires the surface to be closed and have a smooth boundary. Non-smooth or non-orientable surfaces do not satisfy the conditions of the theorem and cannot be used.

5. How is Stoke's theorem of hemisphere related to other mathematical theorems?

Stoke's theorem of hemisphere is closely related to the more general Stokes' theorem, which applies to any smooth surface in three-dimensional space. It is also a special case of the more advanced Gauss-Bonnet theorem, which relates the curvature of a surface to its topology.

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