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skrat
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Homework Statement
Integrate vector field ##\vec{F}=(x+y,y+z,z+x)## over surface ##S##, where ##S## is defined as cylinder ##x^2+y^2=1## (without the bottom or top) for ##z\in \left [ 0,h \right ]## where ##h>0##
Homework Equations
The Attempt at a Solution
Since cylinder is not closed (bottom and top are not included) I firstly have to take this into account, therefore Gaussian law is:
##\int \int _{S}+\int \int _{O}=\int \int \int _{Cylinder}\nabla\vec{F}dV##
so
##\int \int \int _{Cylinder}3dV=3\pi h##
Now, to calculate the vector filed over surface S, we need to firstly calculate the vector field over top and bottom (surface O).
Polar coordinates ##x=r\cos\phi ## and ##y=r\sin\phi ## for ##r\in \left [ 0,1 \right ]## and ##\phi \in \left [ 0,2\pi \right ]##.
Than ##\vec{F}(\phi , r)=(r\cos\phi + r\sin\phi , r\sin\phi + h,h+r\cos\phi )## and ##r_{phi}\times r_r=(0,0,-r)##
Than for bottom where ##z=0##: (Direction of ##r_{phi}\times r_r## is ok)
##\int_{0}^{2\pi }\int_{0}^{1}-r^2cos\phi drd\phi =0##
but for ##z=1##: (Direction of ##r_{phi}\times r_r## sign needs to be corrected)
##\int_{0}^{2\pi }\int_{0}^{1}(rh+r^2cos\phi ) drd\phi =\pi h##
So ##\int \int _{S}=2\pi h##
Or is my idea of surface orientation completely messed up?