# what is the mistake here

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• Jun 30th 2011, 02:46 PM
transgalactic
what is the mistake here
why does the prof wrote a mistake there and wrote 2cos theta in there
??

(i didnt typed the question correctly,the original is
i need to calculate the integral on the volume enclosed by z>=0 and the sphere wich is written in the photo.)
http://i52.tinypic.com/24oq2ww.jpg
??
• Jun 30th 2011, 05:50 PM
ojones
Re: what is the mistake here
Your prof is correct; you need to substitute spherical coordinates into the equation for the volume. Your limits are wrong because you are not integrating over a sphere of radius 2 centered at the origin.
• Jul 1st 2011, 12:52 AM
transgalactic
Re: what is the mistake here
correct i am integration on sphere centered (0,0,1)

in spherecal coordinated the third coordinate is the radius and from the photo and you said it youself that the radius is 2.

when i change the coordinated from cartesian to spherecal
i do it from the sketch,so i get radius from 0 till 2
cant see how the shifted center changes the intervals the way it did.
• Jul 1st 2011, 01:36 AM
ojones
Re: what is the mistake here
The spherical coordinates are with respect to the Cartesian axes, not the center of the sphere that you're integrating over. Substituting the expressions for the spherical coordinates into $\displaystyle x^2+y^2+z^2\le 2z$ gives $\displaystyle \rho\le 2\cos \phi$.

You can shift the Cartesian coordinates to the center of the sphere but then you need to make the corresponding change to the integrand.
• Jul 1st 2011, 01:55 AM
transgalactic
Re: what is the mistake here
Quote:

Originally Posted by ojones
The spherical coordinates are with respect to the Cartesian axes, not the center of the sphere that you're integrating over. Substituting the expressions for the spherical coordinates into $\displaystyle x^2+y^2+z^2\le 2z$ gives $\displaystyle \rho\le 2\cos \phi$.

You can shift the Cartesian coordinates to the center of the sphere but then you need to make the corresponding change to the integrand.

ok ,but how mathematicky can i get this transition
how to get this expression
$\displaystyle \rho\le 2\cos \phi$
• Jul 1st 2011, 03:13 PM
ojones
Re: what is the mistake here
You did something similar already on your exam; how else did you transform the integrand to $\displaystyle 1/\rho$?
• Jul 1st 2011, 03:47 PM
transgalactic
Re: what is the mistake here
Quote:

Originally Posted by ojones
You did something similar already on your exam; how else did you transform the integrand to $\displaystyle 1/\rho$?

why would i want
$\displaystyle 1/\rho$
?
• Jul 1st 2011, 03:50 PM
ojones
Re: what is the mistake here
My point is how did you get $\displaystyle 1/\rho$?
• Jul 1st 2011, 03:57 PM
transgalactic
Re: what is the mistake here
Quote:

Originally Posted by ojones
My point is how did you get $\displaystyle 1/\rho$?

$\displaystyle \rho^2=x^2+y^2+z^2$
so i got it from putting the $\displaystyle \rho$ instead of the expression in the denominator
cant see how it changes the interval into this cosine on top
?
• Jul 1st 2011, 04:00 PM
ojones
Re: what is the mistake here
You skipped a big step here. How is this identity obtained?
• Jul 1st 2011, 04:07 PM
transgalactic
Re: what is the mistake here
between spherecal and cartesian
but we dont use them
we just see the sketch and build the integral by it.

can you tell me from the transision formulas
why $\displaystyle \rho$
have these intervals?
• Jul 1st 2011, 04:41 PM
ojones
Re: what is the mistake here
You don't use the expressions relating spherical to cartesian coordinates?? That's absurd. This is why you don't understand your professor's comments. Go back to the beginning of this subject and learn it properly this time. There are no shortcuts, believe me.
• Jul 2nd 2011, 12:46 AM
transgalactic
Re: what is the mistake here
between spherecal and cartesian
but we dont use them
we just see the sketch and build the integral by it.

can you tell me from the transision formulas
why $\displaystyle \pho$
have these intervals?

you dont use these formulas to build the integral
you do it by the sketch

but here i dont know how to get the 2cosine interval
could you show me the transition?
• Jul 2nd 2011, 01:05 AM
ojones
Re: what is the mistake here
I'll give you a hint. What is $\displaystyle z$ in spherical coordinates?
• Jul 2nd 2011, 01:14 AM
transgalactic
Re: what is the mistake here
$\displaystyle z=\rho cos \theta$
what to do now?
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