Question

F-dr = 47 for a vector function F = (2x -y)i A line integral is evaluated as yzz -y?zk acting on the curved surface S, where
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Answer #1

Solution:

k a V x F 2x - y -y y

Now, the surface integral over the surface \small S of the hemisphere is same as the surface integral over  \small S_{1} , the plane face of the hemisphere as both have the same boundary over  \small C .   

Further, in  \small xy-plane N k, ds dr dy

x F) ds N (

k k dr dy

dr dy S

Area of S

where, \small S_{1} is the area of the circle  y which is  \small 4\pi .

\small \int \int _{C}N\cdot \left ( \bigtriangledown \times F \right )ds=4\pi

Thus, the stoke theorem is verified.

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