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Surface Integral of Vector Field 
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#1
Nov2408, 12:23 PM

P: 4

1. The problem statement, all variables and given/known data
Find [tex]\int\int_{S}[/tex] F dS where S is determined by z=0, 0[tex]\leq[/tex]x[tex]\leq[/tex]1, 0[tex]\leq[/tex]y[tex]\leq[/tex]1 and F (x,y,z) = xi+x^{2}jyzk. 2. Relevant equations T_{u}=[tex]\frac{\partial(x)}{\partial(u)}[/tex](u,v)i+[tex]\frac{\partial(y)}{\partial(u)}[/tex](u,v)j+[tex]\frac{\partial(z)}{\partial(u)}[/tex](u,v)k T_{v}=[tex]\frac{\partial(x)}{\partial(v)}[/tex](u,v)i+[tex]\frac{\partial(y)}{\partial(v)}[/tex](u,v)j+[tex]\frac{\partial(z)}{\partial(v)}[/tex](u,v)k [tex]\int\int_{\Phi}[/tex] F dS = [tex]\int\int_{D}[/tex] F * (T_{u}xT_{v}) du dv 3. The attempt at a solution To start off, I'm not sure how to parametrize the surface S. Any help is appreciated. 


#2
Nov2408, 01:47 PM

Math
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Thanks
PF Gold
P: 39,568

Since you are just talking about a portion of the xyplane, x= u, y= v, z= 0. Oh, and the order of multiplication in [itex]T_u\times T_v[/itex] is important. What is the orientation of the surface? (Which way is the normal vector pointing?)
(Actually that last point doesn't matter because this integral is so trivial.) 


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