Is 1 + e^(-x^2+y^2) a two-sheeted hyperboloid?
It's just an expression so far. Do you mean z=1+e^(-x^2+y^2), that's a surface.
sorry, yes. and I made a typo--it's z= 1+ e^(-x^2-y^2).
It's not a hyperboloid. A hyperboloid is a quadratic form. That's not, it has an e^ in it. And it doesn't have two sheets. In cylindrical coordinates it's z=1+e^(-r^2). Can you picture what that looks like?
I don't understand what you mean by cylindrical coordinates, I've never heard that term. I have trouble picturing things, too, until I start plotting points. Is that what I should do, just begin plotting points until I start to see it?
Better to think about it a bit before you start plotting points. In z=1+e^(-r^2) I'm taking r=sqrt(x^2+y^2). So r is just the distance from (0,0) to (x,y) in the x-y plane. Can you picture it now?
I can't quite see it yet. why wouldn't you just let r=-x^2-y^2?
Your choice. I usually like to pick r>=0. But look, at (x,y)=(0,0), z=2. As the distance of (x,y) from (0,0) gets larger and larger, -x^2-y^2 gets larger and larger in a negative way. So e^(-x^2-y^2) get closer to 0. So z->1 at infinity.
ok, I see how you get z=2, at least. I feel so stupid in that I only get about half of the second part and I absolutely cannot see what this thing looks like! I'm thinking of z as a level--is that wrong? looking at this from the xy plane, does it look like a function like (1/x)^2?
I have nothing in my notes about any surfaces that have e in them, nor are there any examples like this in the text--that's why I'm so lost.
'z' is the height above (or below) the x-y plane for a given value of x and y. Maybe plotting some points isn't that bad an idea. Try some.
and now comes the inane question about how to plot some points. just pick random x's and y's?
oh wait, I just need to choose a z and use logarithms to find y in terms of x. correct?
You are asking before you are thinking about it. How about (x,y)=(1,0),(2,0),(3,0) etc. Then (x,y)=(0,1),(0,2),(0,3)... Or (x,y)=(1,1),(2,2),(3,3).... You don't actually have to plot them all, just think about what would happen if you did.
No, you actually have to think about what the surface would look like if you did plot a bunch of points. This isn't that hard.
when x and y grow larger, z approaches 1. that is why I thought it looked like (1/x)^2, because that decreases exponentially towards an asymptote. the difference is that this, instead of being a line is a surface. correct?
oh, I know it shouldn't be that hard, but I really appreciate your coaching me through this. just think where I'd be without you!
What it really looks like is 1+e^(-x^2). But, yes, certainly, it decreases exponentially towards an asymptote. It doesn't have two sheets and it's not a hyperboloid.
I tremble to think. :)
ok. so if I were to draw this thing, it would basically look like a flat plane with a hump in it near the origin? that's what brought on this question; I'm supposed to be drawing it.
Yes, an almost flat sheet at infinity with a hump at the origin.
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