Double of F*dS,over S, where
and where S is the surface of the infinite cylinder x^2+y^2=1; n pointing outward© BrainMass Inc. brainmass.com September 28, 2022, 3:27 am ad1c9bdddf
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Instead of dS I used the notation da, since ds is usually used for path integral, rather surface integral.
We can write the surface unit as
The normal vector is:
And the surface element is:
On the surface of the cylinder r=1 thus:
The vector field is
Converting to cylindrical coordinates:
Using again the fact that on the surface r=1 we get:
For the integral we have between 0 and 2, and z between -∞ to ∞.
The integral becomes:
Using the substitution we obtain:
But , so:© BrainMass Inc. brainmass.com September 28, 2022, 3:27 am ad1c9bdddf>