Proving radial properties of particular dimensionless surface plots?

[tade]

We have a surface function z = f(x,y) ; f(x,y) only contains dimensionless constants, and is itself dimensionless.

If we convert it to cylindrical co-ordinates, z = f(r,θ) , does z only depend on θ?
Meaning we can remove r from the equation, literally.

 

[BvU]

z only depend on θ?

Obviously not ! Look at z(r) for e.g. $\theta=\pi/4$

 

[tade]

Obviously not ! Look at z(r) for e.g. $\theta=\pi/4$

what's the equation of f(r,θ) ?

 

[BvU]

You are asking me ?
Or do you mean: if $\theta=pi/4$ then $f(r,\theta) = f(x,x)$ with $x=r/\sqrt 2$ ?

 

[tade]

You are asking me ?

I thought you had come up with a counter-example to prove the conjecture false, so I was wondering what that counter-example function was.

 

[BvU]

Conjecture: z is independent of r
Counter example: along the diagonal I see z go up, down, up again and then down again -- clearly not independent of r

 

[tade]

Conjecture: z is independent of r
Counter example: along the diagonal I see z go up, down, up again and then down again -- clearly not independent of r

sorry, what's the equation of the counter-example function?

 

[BvU]

However, I see an interpretation of

f(x,y) only contains dimensionless constants

If x and y do not occur, then f itself is a constant, therefore independent of r, but equall indepndent of $\theta$

 

[tade]

However, I see an interpretation of
If x and y do not occur, then f itself is a constant, therefore independent of r, but equall indepndent of $\theta$

Oh, so just z = c , the surface plot being just a flat plane?
That's a trivial case and the conjecture is that there'll be r-independence in all cases, whichever function you use. As long as the function satisfies the two conditions.

 

[BvU]

In which case your picture is wrongfooting any good-willing helper

 

[tade]

In which case your picture is wrongfooting any good-willing helper

Well, it's certainly not an r-independent function, though I just wanted to make sure people got the idea of "surface plot" immediately.

 

[BvU]

As long as the function satisfies the two conditions

I don't see what the dimensionlessness of f or its contained constants has to do with it

 

[BvU]

just wanted to make sure people got the idea of "surface plot" immediately

Well, this exercise creates more confusion than it removes

 

[tade]

Well, this exercise creates more confusion than it removes

Well, no equations are given, the image not referenced, cos its a general conjecture.

I should add some details in the OP though, but its too late now.

 

[tade]

I don't see what the dimensionlessness of f or its contained constants has to do with it

Its part of the conjecture. Unless its possible to expand the generality of the conjecture even further.