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Contour Curves

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
Hi!!!
Let the function be f(x,y)=x^2-2*y^2,which graph is S:z=f(x,y).Which are the contour curves???Are these hyperbolas??? :confused:

Thanks in advance!!!!:)
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
put \(\displaystyle x^2-y^2= k \)

Now , choose some values for $k$ and draw them in the xy-plane . What are these curves ?
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
I did this,and I think that the curves are hyperbolas...Is this correct???

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Or are these hyperbolic paraboloids???
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,887
I did this,and I think that the curves are hyperbolas...Is this correct???

- - - Updated - - -

Or are these hyperbolic paraboloids???
Heh. The surface is a hyperbolic paraboloid.
The contour curves are indeed hyperbolas.
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
Nice...Thank you very much!!! :p
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
I have also an other question... :eek: To find the tangent plane at the point (sqrt(2),1,0),I have to find df/dx,df/dy,df/dz...Is df/dz=d(x^2-2*y62-z)/dz=-1???Or df/dz=dz/dz=1???
 

Petrus

Well-known member
Feb 21, 2013
739
I have also an other question... :eek: To find the tangent plane at the point (sqrt(2),1,0),I have to find df/dx,df/dy,df/dz...Is df/dz=d(x^2-2*y62-z)/dz=-1???Or df/dz=dz/dz=1???
An equation of the tangent plane to the surface z = f(x, y) at the point P (
is:


Regards.
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
Great...!!!Thank you very much...!!! :rolleyes:
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,887
I have also an other question... :eek: To find the tangent plane at the point (sqrt(2),1,0),I have to find df/dx,df/dy,df/dz...Is df/dz=d(x^2-2*y62-z)/dz=-1???Or df/dz=dz/dz=1???
Well, this is a bit ambiguous.
The surface z=f(x,y) is a parametric surface in x and y.
It makes no sense to consider df/dz which would be zero, since f(x,y) does not contain z.

So I suspect you're supposed to consider g(x,y,z)=f(x,y)-z.
The equation g(x,y,z)=0 identifies the same surface.
And then yes, dg/dz=-1.
 

evinda

Well-known member
MHB Site Helper
Apr 13, 2013
3,723
Oh good!!!!!Thank you very much!!!! ;)
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,887
You're welcome!!!!!!! ;)