# Surface Integral of Vector Field

by ma3088
Tags: integral, parametrization, surface
 P: 4 1. The problem statement, all variables and given/known data Find $$\int\int_{S}$$ F dS where S is determined by z=0, 0$$\leq$$x$$\leq$$1, 0$$\leq$$y$$\leq$$1 and F (x,y,z) = xi+x2j-yzk. 2. Relevant equations Tu=$$\frac{\partial(x)}{\partial(u)}$$(u,v)i+$$\frac{\partial(y)}{\partial(u)}$$(u,v)j+$$\frac{\partial(z)}{\partial(u)}$$(u,v)k Tv=$$\frac{\partial(x)}{\partial(v)}$$(u,v)i+$$\frac{\partial(y)}{\partial(v)}$$(u,v)j+$$\frac{\partial(z)}{\partial(v)}$$(u,v)k $$\int\int_{\Phi}$$ F dS = $$\int\int_{D}$$ F * (TuxTv) 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.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,553 Since you are just talking about a portion of the xy-plane, x= u, y= v, z= 0. Oh, and the order of multiplication in $T_u\times T_v$ 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.)