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September 9th, 2010, 04:02 PM   #1
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ISSUES with circles...

A circle in R^3, ie 3D space is written as: f(x,y)= x^2+ y^2

If we want to draw level curves we give the function different values of C, for example:
C= 1 --> x^2+ y^2 = 1, This will give us a circle with the center in (0,0) and the radius = sqrt(1)=1

BUT what changes when we have x^2+ 4y^2

What does the 4 change, does it change the centre of the circle?

Also, how should one sketch x^2+y^2+z^2 < 5??

An answe to any of the two questions is highly appreciated...

Please help me!
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September 9th, 2010, 04:21 PM   #2
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Re: ISSUES with circles...

Quote:
Originally Posted by tinyone
What does the 4 change, does it change the centre of the circle?
It turns the circle into an ellipse.
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September 10th, 2010, 02:29 PM   #3
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Re: ISSUES with circles...

Quote:
Originally Posted by tinyone
A circle in R^3, ie 3D space is written as: f(x,y)= x^2+ y^2
Wrong. If you have 3D space, you have a function f(x,y,z). You have R^3 -> R^2. z is not used in the function but it does exist.

Quote:
If we want to draw level curves we give the function different values of C, for example:
C= 1 --> x^2+ y^2 = 1, This will give us a circle with the center in (0,0) and the radius = sqrt(1)=1
wrong, see above.
Quote:
BUT what changes when we have x^2+ 4y^2
What does the 4 change, does it change the centre of the circle?
See the last post.

Quote:
Also, how should one sketch x^2+y^2+z^2 < 5??
Now we are going from 3D to 3D since you do not determine it to a fixed number ... this is an open sphere, with a volume.
You seem to misunderstand functions. Think about what R^n -> R^m means, as a hint.
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September 10th, 2010, 06:43 PM   #4
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Re: ISSUES with circles...

ok good point

but is it not possible to draw level curves for f(x,y)= x^2+ y^2 by assigning it different constants?
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September 11th, 2010, 03:42 AM   #5
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Re: ISSUES with circles...

Quote:
Originally Posted by tinyone
. . . but is it not possible to draw level curves for f(x,y)= x^2+ y^2 by assigning it different constants?
Yes. If z = f(x, y) = x^2 + y^2 you have an elliptic paraboloid.

Quote:
Originally Posted by tinyone
x^2+ 4y^2
In space, this is an elliptic paraboloid. In the plane it is an ellipse.
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September 11th, 2010, 09:00 AM   #6
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Re: ISSUES with circles...

Thank you very much greg1313, that cleared up a lot!
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