Use Stokes' theorem for both parts, which equates the surface integral of the curl to the line integral along the surface's boundary.
a. The boundary of the hemisphere is the circle in the plane
, where the curl is
. Green's theorem applies here, so that
which means the value of the line integral is 3 times the area of the circle, or .
b. The closed sphere has no boundary, so by Stokes' theorem the integral is 0.
Most of the time for coordinate planes, they are asking you to plot the point correctly, and to state which quadrant the point falls into.
The Quadrant points start at (+ , +) for Quadrant I, and moves clockwise. See attached picture.
Quadrant I: (+ , +)
Quadrant II: (+ , -)
Quadrant III: (- , -)
Quadrant IV: (- , +)
Remember, the point is split into two parts, in which the first one is x (which signifies the placement for the horizontal line), while the other is y (which signifies the placement for the vertical line).
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