Is it possible for two different level curves of a function to intersect?
if you have the function z=f(x,y) and the different level curves \[ K_\alpha=\{(x,y)\in\mathbb{R}^2:\alpha=f(x,y)\} \] and \[ K_\beta= \{(x,y)\in\mathbb{R}^2:\beta=f(x.y)\}\] \[ \alpha\neq\beta\] If \[ K_\alpha\cap K_\beta\neq\emptyset \] what would that mean?
\[K_\alpha=\{(x,y)\in\mathbb{R}^2:\alpha=f(x,y)\}\] I'm not very good at understanding mathematical notation, what do we mean here?
a level curve if the set of point (x,y) all of which have the same image under f. that's what K_alpha means
I'm embarrassed to say I still don't understand
imagine you have an orange,ok
ok
now if you cut the orange at two different hights (say from the bottom) what would you see?
if you look at the orange from above
|dw:1341630949923:dw| Like this?
exactly the knife is like a horizontal plane
when you "cut" a 3d graph with a horizontal plane (knife) you get diferent 2d figures
no say you cut 2 slices of the orange at different hights would they be equal or different?
different
unless it's a cylnder, it would be different
the lines you drew are level curves do they intersect?
no they don't intersect
Ok. you got it. In the case of a vertical cilinder ALL level curves would be the same right?
yes
but in the case of a paraboloid (i don't know it this is the right word in english) with vertex at (0,0,0) all level curves would be different right?
yes
and the case of the orange (or a sphere) some level curves would be different but some would be equal right?
yes
so you can't say for sure if two different level curves will intersect or not
Why couldn't we be certain since their at different levels and wouldn't intersect
because the set itself is not 3-dimensional it is 2-dimensional like what i wrote at the begining
give a second
ok
i hope this helps
yep I'm reading it right now
here is another one
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