Thread: do Carmo question - differential geometry....

Let P = {(x,y,z) : x=y}, and let a function x:U->R^3 be given by x(u,v) = (u+v,u+v, uv) where U = {(u,v) : u>v}

Is x a parameterization of P?

I'm really stuck on this question and I feel it is easier than I am making it out to be.

The answer is Yes, (as it says this in the back of the book) - i'm not really sure how to be rigorous in proving it though.

I see that in the function x, x = y. so this is a start. I think What the question is asking is that given any (x,y,z) can you find a u and v such that x = y = u+v and uv = z??

I have got something like x = u + v, z = uv and then have solved for u and v in terms of x and z. Is this all we need to show? I end up getting u and v are derived from the quadratic formula.



the same for v...

this would mean that x^2 needs to be > 4z... does this matter?

now if u>vwould mean you would need to take the positive sqare root for u and negative for v. is this right?

If u = v, then it is possible we might run in to problems.

Please help someone... im so shite at all this

The point belongs to , however the system has no real solutions. Perhaps there is a typo in your book.

Originally Posted by mathswannabe

Let P = {(x,y,z) : x=y}, and let a function x:U->R^3 be given by x(u,v) = (u+v,u+v, uv) where U = {(u,v) : u>v}

Is x a parameterization of P?

I'm really stuck on this question and I feel it is easier than I am making it out to be.

The answer is Yes, (as it says this in the back of the book) - i'm not really sure how to be rigorous in proving it though.

I see that in the function x, x = y. so this is a start. I think What the question is asking is that given any (x,y,z) can you find a u and v such that x = y = u+v and uv = z??

I have got something like x = u + v, z = uv and then have solved for u and v in terms of x and z. Is this all we need to show? I end up getting u and v are derived from the quadratic formula.



the same for v...

this would mean that x^2 needs to be > 4z... does this matter?

now if u>vwould mean you would need to take the positive sqare root for u and negative for v. is this right?

If u = v, then it is possible we might run in to problems.

Please help someone... im so shite at all this

I feel as though this might be helpful.

Originally Posted by Drexel28

I feel as though this might be helpful.

So, we need two books.