Surface Integral of Vector Field

[ma3088]

Homework Statement

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.

Homework Equations

T_u=$$\frac{\partial(x)}{\partial(u)}$$(u,v)i+$$\frac{\partial(y)}{\partial(u)}$$(u,v)j+$$\frac{\partial(z)}{\partial(u)}$$(u,v)k

T_v=$$\frac{\partial(x)}{\partial(v)}$$(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 * (T_uxT_v) du dv

The Attempt at a Solution

To start off, I'm not sure how to parametrize the surface S. Any help is appreciated.

 

[HallsofIvy]

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.)