Boundary Curve and Stokes Theorem in a Partially Missing Cube

[riemannsigma]

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

 

[The Bill]

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.

 

[riemannsigma]

Thanks