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Mathematics 54 Online
OpenStudy (anonymous):

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

OpenStudy (helder_edwin):

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?

OpenStudy (anonymous):

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

OpenStudy (helder_edwin):

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

OpenStudy (anonymous):

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

OpenStudy (helder_edwin):

imagine you have an orange,ok

OpenStudy (anonymous):

ok

OpenStudy (helder_edwin):

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

OpenStudy (helder_edwin):

if you look at the orange from above

OpenStudy (anonymous):

|dw:1341630949923:dw| Like this?

OpenStudy (helder_edwin):

exactly the knife is like a horizontal plane

OpenStudy (helder_edwin):

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

OpenStudy (helder_edwin):

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

OpenStudy (anonymous):

different

OpenStudy (anonymous):

unless it's a cylnder, it would be different

OpenStudy (helder_edwin):

the lines you drew are level curves do they intersect?

OpenStudy (anonymous):

no they don't intersect

OpenStudy (helder_edwin):

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

OpenStudy (anonymous):

yes

OpenStudy (helder_edwin):

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?

OpenStudy (anonymous):

yes

OpenStudy (helder_edwin):

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

OpenStudy (anonymous):

yes

OpenStudy (helder_edwin):

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

OpenStudy (anonymous):

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

OpenStudy (helder_edwin):

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

OpenStudy (helder_edwin):

give a second

OpenStudy (anonymous):

ok

OpenStudy (helder_edwin):

i hope this helps

OpenStudy (anonymous):

yep I'm reading it right now

OpenStudy (helder_edwin):

here is another one

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