- #1
Lucy Yeats
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Homework Statement
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.
Homework Equations
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! :-)