Question

Is it possible for two different level curves of a function to intersect?

Answer 1

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?

Answer 2

\[K_\alpha=\{(x,y)\in\mathbb{R}^2:\alpha=f(x,y)\}\]
I'm not very good at understanding mathematical notation, what do we mean here?

Answer 3

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

Answer 4

I'm embarrassed to say I still don't understand

Answer 5

imagine you have an orange,ok

Answer 6

ok

Answer 7

now if you cut the orange at two different hights (say from the bottom)
what would you see?

Answer 8

if you look at the orange from above

Answer 9

Like this?

Answer 10

exactly
the knife is like a horizontal plane

Answer 11

when you "cut" a 3d graph with a horizontal plane (knife) you get diferent 2d figures

Answer 12

no say you cut 2 slices of the orange at different hights
would they be equal or different?

Answer 13

different

Answer 14

unless it's a cylnder, it would be different

Answer 15

the lines you drew are level curves
do they intersect?

Answer 16

no they don't intersect

Answer 17

Ok.
you got it.
In the case of a vertical cilinder ALL level curves would be the same
right?

Answer 18

yes

Answer 19

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?

Answer 20

yes

Answer 21

and the case of the orange (or a sphere) some level curves would be different but some would be equal
right?

Answer 22

yes

Answer 23

so you can't say for sure if two different level curves will intersect or not

Answer 24

Why couldn't we be certain since their at different levels and wouldn't intersect

Answer 25

because the set itself is not 3-dimensional it is 2-dimensional like what i wrote at the begining

Answer 26

give a second

Answer 27

ok

Answer 28

i hope this helps

Answer 29

yep I'm reading it right now

Answer 30

here is another one