Evaluating Surface Integral with Stokes Theorem

In summary, Stokes' theorem is used to evaluate the surface integral of the curl of a vector field F dot dS, where F is equal to (x^2 + y^2, x, 2xyz) and S is an open surface x^2 + y^2 + z^2 = a^2 for z >= 0, which is a hemisphere of radius a lying on the x-y plane. The closed curve dr bounding this hemisphere is the circle x² + y² = a² in the xy plane. The limits of integration can be found by converting to spherical polar coordinates and differentiating. Drawing a picture can also help visualize the boundary of the surface.
  • #1
trelek2
88
0
use the stokes theorem to evaluate the surface integral [curl F dot dS] where
F=(x^2+y^2; x; 2xyz)
and S is an open surface x^2+y^2+z^2=a^2 for z>=0. So i guess its a hemisphere of radius a lying on x-y plane.
I don't see however how to take F dot dr. What is this closed curve dr bounding this hemisphere? I guess we can take spherical polar coordinates, but still once i have x,y,z in terms of r,phi,theta i still don't know "dr" (just differentiate ?) and what will the limits of integration be? Can someone show me?
 
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  • #2
The boundary of the surface is just the circle x² + y² = a² in the xy plane. Did you try drawing a picture?
 

1. What is the purpose of using Stokes Theorem in evaluating surface integrals?

Stokes Theorem is a powerful tool that allows us to evaluate surface integrals by converting them into line integrals. This makes the calculation easier and more efficient, as line integrals are typically easier to solve. It also provides a connection between the surface and its boundary, making it useful in many applications in physics and engineering.

2. How does Stokes Theorem work?

Stokes Theorem states that the surface integral of a vector field over a closed surface is equal to the line integral of the same vector field over the boundary of the surface. This means that the flux of the vector field through the surface can be calculated by integrating the field along the boundary curve.

3. What are the conditions for using Stokes Theorem?

The conditions for using Stokes Theorem are that the surface must be smooth and orientable, and the vector field must be continuous and differentiable on the surface and its boundary. The surface must also be closed, meaning that it has no boundary.

4. Can Stokes Theorem be used to evaluate any surface integral?

No, Stokes Theorem can only be used to evaluate surface integrals where the surface is a closed, smooth, and orientable surface. Additionally, the vector field must be continuous and differentiable on the surface and its boundary. If these conditions are not met, other methods of integration must be used.

5. What are some practical applications of Stokes Theorem?

Stokes Theorem has many practical applications in physics and engineering. It is commonly used in fluid mechanics to calculate the circulation of a fluid around a closed loop. It is also used in electromagnetism to calculate the work done by a magnetic field on a moving charged particle. In addition, Stokes Theorem is useful in calculating surface integrals in three-dimensional space, which has applications in fields such as computer graphics and geology.

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