Contour line of Injective function

[gipc]

Hello,

I have g(t) is a continuous and a differential function under 1 variable.

let h(x,y)=g(x^2+y^2)

suppose that g(t) is Injective (thus monotonous)

What is the shape of the contour lines of the graph of h(x,y)?

-I have a sense that we're talking about simple cycles but I don't know how to show it.
should i use the inverse function g^-1(t)or something?

And secondly, what would the contour lines look like if g wasn't monotonous? I for one have no idea and I would appreciate some help.

 

[tiny-tim]

hello gipc!

(try using the X² icon just above the Reply box )

it's asking what are the curves h(x,y) = constant …

does that help? ​

 

[gipc]

Yes, i got this- the contour lines are circles when the function is monotonous.

The question is when the function isn't monotonous, how does that affect the contour lines? How do they change?

 

[Tedjn]

When g is not injective, g^-1(z) might contain more than one element. How would that differ in picture from when the inverse image contains only one element? I think you can figure this out.

 

[tiny-tim]

hi gipc!

monotone: each contour line is a circle

not monotone: each contour line is one or more circles