Sketching level curves

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December 9th 2011, 04:37 AM
SyNtHeSiS

Sketching level curves

Sketch the level curves for the function $z = ln(x^2 + y^2)$ .

Attempt:

$k = ln(x^2 + y^2)$

$z = ln(k^2 + y^2)$

$z = ln(x^2 + k^2)$

Now I am not sure how I would draw each of these graphs due to the $ln(x^2 + y^2)$ part. (Worried)

Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)
December 9th 2011, 07:10 AM
alexmahone

Re: Sketching level curves

Quote:

Originally Posted by SyNtHeSiS

Sketch the level curves for the function $z = ln(x^2 + y^2)$ .

Attempt:

$k = ln(x^2 + y^2)$

$z = ln(k^2 + y^2)$

$z = ln(x^2 + k^2)$

Now I am not sure how I would draw each of these graphs due to the $ln(x^2 + y^2)$ part. (Worried)

Also when asked to sketch the level curves, does that mean you just let z = k and draw one graph? (as opposed to also letting x = k and y = k)

$z=k$

$\ln (x^2+y^2)=k$

$x^2+y^2=e^k$

So, the level curves are circles in the xy-plane that are centered at the origin.
December 10th 2011, 12:17 AM
SyNtHeSiS

1 Attachment(s)

Re: Sketching level curves

Thanks :). Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for $x = k$ and $y = k,$ before drawing $x = ln(x^2 + y^2))$ ?
December 10th 2011, 12:23 AM
alexmahone

Re: Sketching level curves

Quote:

Originally Posted by SyNtHeSiS

Thanks :). Would drawing the graph give you a circular paraboloid (or do you have to draw the traces for $x = k$ and $y = k,$ before drawing $x = ln(x^2 + y^2))$ ?

No, because the equation of a circular paraboloid is $x^2+y^2=a^2z$ .
December 16th 2011, 01:51 AM
SyNtHeSiS

Re: Sketching level curves

Oh ok. What steps would you use to draw the correct graph?
December 16th 2011, 01:58 AM
alexmahone

1 Attachment(s)

Re: Sketching level curves

Quote:

Originally Posted by SyNtHeSiS

Oh ok. What steps would you use to draw the correct graph?

$z=\ln (x^2+y^2)=\ln r^2=2\ln r$

$r=e^{z/2}$

I have attached a graph generated using MATLAB.
December 16th 2011, 06:11 AM
HallsofIvy

Re: Sketching level curves

Any function of the form $z= f(x^2+ y^2)= f(r)$ has circular symmetry. In particular, $z= ln(x^2+ y^2)= ln(r^2)= 2ln(r)$ is a logarithm graph. Draw that and rotate around the z-axis.