Surface Integral of Vector Field

[ma3088]

Homework Statement

Find \int\int_{S} F dS where S is determined by z=0, 0\leqx\leq1, 0\leqy\leq1 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.)