Let
(a)
ANSWER:
(b) Let be the circle in -plane, oriented clockwise. So, is a closed path in -plane.
By Stoke's theorem, since where is the surface on -plane bounded by the circle .
(c) Let be any closed curve in -plane. Again by Stoke's theorem since where is the surface bounded by the closed curve .
(d) Let be the half-circle in -plane with , transversed from to . Let's first draw and show it's direction/orientation.
Now, let denotes the line joining the endpoints of i.e. and , traversed from to .
In other words
Let's plot draw in on the graph.
Now, observe and together constitute a closed curve on -plane traversing counter clockwise. We call .
From the part (c), we know that as is a closed curve.
Now,
So if we can compute , we can get . So, let's find .
By definition, where and
So, and .
Thus
Hence,
So, from we get,
ANSWER:
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