I didn't check your detailed computation but yes the argument is correct.
For which values of , is a regular surface , where
is defined as
I found 2 theorems I think I should use when solving this problem
thm 1
is a regular value of the function iff the partial derivatives do not disappear instantly for any point of the pre-image
thm 2
If is differentiable and is a regular value , then is a regular surface
So I start off with finding critical points (point where all partial derivatives disappear at the same time)
I get the following points:
so that means that all point except the above ones are regular points?
the second theorem gives that is a regular surface
Is this solution correct??
Thank you in advance
thank you for your reply.
I would appreciate if you can have a look at my other post where I am trying to show that a function is smooth
smooth function