Use Stokes' Theorem to evaluate C F · dr F(x, y, z) = xyi + yzj + zxk, C is the boundary of the part of the paraboloid z = 1 − x
2 − y2 in the first octant.
1 answer:
I assume
has counterclockwise orientation when viewed from above.
By Stokes' theorem,

so we first compute the curl:


Then parameterize
by

where the
-component is obtained from

with
and
.
Take the normal vector to
to be

Then the line integral is equal in value to the surface integral,



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