Type your question here Suppose a student uses Stokes theorem and computes times

Question

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Suppose a student uses Stokes theorem and computes times F.ndS over the upper half of a sphere centered at the origion and obtains some value, which we will call C. the student then wishes to compute the integral times F.ndS over the bottom half of the same sphere and reasons that, since the two hemispheres have the same boundary curve, with different orientations, the value of the integral over the bottom portion of the sphere is - C. Explain this is not necessarily true, and give an example of a vector field F where this would be true.

Explanation / Answer

Let S be the upper hemisphere of the unit sphere x^2 + y^2 + z^2 = 1.

If the vector field be F = x i + y j + z k,

then the above will be true, as when the line integral F.ds is performed over the boundary curve C, for the upper hemisphere C would be counter clockwise and for the lower hemisphere it would be clockwise, thus yielding same value but with opposite signs.

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