(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Evaluate the surface integral ∫F.dS where F = xi - yj + zk and where the surface S is of the cylinder defined by x^2+y^2≤4, and 0≤z≤1. Verify your answer using the Divergence Theorem.

2. Relevant equations

3. The attempt at a solution

I parametrized the surface in terms of θ and z: r=(2cosθ, 2sinθ, z). I found dr/dθ X dr/dz=(2cosθ, 2sinθ, z). (How do I know which way round to do the cross product?). I rewrote F as (2cosθ, -2sinθ, z). I then did the following integral:

∫∫(2cosθ, -2sinθ, z).(2cosθ, 2sinθ, z)dθdz and got ∫∫4(cosθ)^2-4(sinθ)^2 dθ dz=0

But using the divergence theorem: divF=1, so I found the answer to be the volume of the cylinder, 4pi.

Where have I gone wrong?

Thanks in advance! :-)

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# Homework Help: Where have I gone wrong with this surface integral problem?

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