# Surface Integral of Vector Field

1. Nov 24, 2008

### ma3088

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.

2. Nov 24, 2008

### HallsofIvy

Staff Emeritus
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.)