Analyzing surfaces and curves using Implicit Function Thm

[trap101]

for each of the following maps f: ℝ²-->ℝ³, describe the surface S = f(ℝ²) and find a description of S as the locus of an equation F(x,y,z) = 0. Find the points where \partial_uf and \partial_vf are linearly dependent and describe the singularities of S(if any) at these points

f(u,v) = (2u + v, u-v , 3u) (parametric equation)


Attempt:

So what I do know how to do is determine whether there are points where the partials are linearly dependent, that's done by taking the cross product.

But the initial part is where I'm having problems. In the solutions, they were able to describe this as S is the plane x+y = z. Now from just finishing a few problems between R^1 and R^2, the only way I was mainly able to figure out a description of the shape was through some plugging in and drawing of the graph. Is this what I would have to do for these questions as well? Also what kind of description are they looking for in terms of the locus? and finally when they ask about the singularity of S at a point are they just talking about where the partials are not linearly independent?

 

[HallsofIvy]

It's really just a matter of solving three equations in three unknown values. You are given that x= 2u+ v, y= u- v, and z= 3u. Adding the first two equations eliminates v and gives x+y= 3u so that u= (x+ y)/3. Putting that into z= 3u, z= 3(x+ y)/3= x+ y. That is, z= x+y so F(x, y, z)= x+y- z= 0.