Boundary Curve and Stokes Theorem in a Partially Missing Cube

In summary, the conversation discusses a 5-sided cube with a missing bottom face and the formation of a boundary curve around the remaining 4 sides. The question is raised about whether STOKE's theorem is still applicable in this scenario, considering the missing bottom face. The speaker clarifies that the boundary of the cube is actually the bottom square and explains how dividing the shape into two parts can resolve any confusion.
  • #1
riemannsigma
10
0
Let's say there is a 5 sided cube that is missing the bottom face.

Obviously, there is a boundary curve at the middle of this cube that goes around the 4 sides, front, right, back, and left.

This boundary curve forms the boundary of the top half of the cube with the 5 faces and the bottom half of the cube with the 4 faces(bottom face is missing)

Does STOKE's theorem break apart here? The curl of the field dot the 5 faces of the cube ought to equal the closed boundary line integral... But I am missing the bottom face of the cube.

HELP
 
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  • #2
It seems as though you are confused about the meaning of "boundary." The boundary of the initial shape you describe is the bottom square, around the missing face.

It seems like you are spitting the shape into two parts in your sentence which begins with "obviously." If so, then the top half has a single square as boundary, and the bottom half has the union of two disjoint squares as its boundary, sharing one with the top half. Taking account of this should resolve the question I think you're asking.
 
  • #3
Thanks
 

What is Stokes theorem?

Stokes theorem is a mathematical theorem that relates the surface integral of a vector field over a closed surface to the line integral of the same vector field along the boundary of the surface.

What is the significance of Stokes theorem?

Stokes theorem is a fundamental tool in vector calculus and has many applications in physics and engineering. It allows for the conversion of a difficult surface integral into an easier line integral, making it a useful tool for solving problems in fluid mechanics, electromagnetism, and more.

What is "Breaking" Stokes theorem?

"Breaking" Stokes theorem refers to a situation where the assumptions of the theorem do not hold, usually due to a discontinuity or singularity in the vector field. In this case, the theorem cannot be applied and alternative methods must be used.

What are the assumptions of Stokes theorem?

The main assumption of Stokes theorem is that the vector field must be continuous and differentiable over the surface and its boundary. Additionally, the surface must be closed and have a smooth boundary.

How is Stokes theorem used in real-world applications?

Stokes theorem has many practical applications, including calculating fluid flow rates in pipes, determining the circulation of air around an airplane wing, and calculating the magnetic field around a wire. It is also used in computer graphics and 3D modeling to create realistic lighting and shading effects.

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