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September 6th, 2017, 10:58 PM  #1 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित  parameterization in Surface integration
I want to surface integrate over a surface of the plane $\displaystyle S : 2x+3y+6z =12$ which lie in the 1st octant. Should I use parameterization, if I should, how?

September 7th, 2017, 03:52 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
A surface, being two dimensional, requires two parameters. You are given 2x+3y+6z=12 which you can write as x= 6 (3/2)y 3z. Take y and z as parameters or, if you prefer, s and t, writing x= 6 (3/2)s 3t, y= s, z= t. Obviously, you can do that with any of the variables: we can just as well write y= 4 (2/3)x 2z and take x= s, y= 4 (2/3)s 2t, z= t or we can write z= 2 (1/3)x (1/2)y and take x= s, y= t, z= 2(1/3)s (1/2)t. Or, if you don't like fractions, take x= 3s, y= 2t, z= 2 s t. 
September 7th, 2017, 05:40 AM  #3 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,829 Thanks: 648 Math Focus: Yet to find out.  
September 7th, 2017, 08:11 AM  #4 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
Yeah, I'm so mean to fractions but they started it!

September 7th, 2017, 08:31 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra 
I think you're being irrational.

September 7th, 2017, 04:52 PM  #6 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,829 Thanks: 648 Math Focus: Yet to find out. 
Get real!

September 11th, 2017, 05:10 AM  #7 
Math Team Joined: Jul 2013 From: काठमाडौं, नेपाल Posts: 901 Thanks: 61 Math Focus: सामान्य गणित 
Suppose i need to integrate a function F = 36z  36 + 18y over the surface, how should i place the limits? 
September 12th, 2017, 04:56 AM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,264 Thanks: 902 
I thought I had already responded to this! In my first post I suggested the parameterization x= 3s, y= 2t, z= 2 s t. Since we can write s and t in terms of x and y separately, we want to project the surface down to the xyplane. The plane 2x+3y+6z=12 crosses the z= 0 plane where 2x+ 3y= 12 and that cuts the xaxis where 2x= 12 or x= 6. To integrate over that, we can take x going from 0 to 6 and, for each x, y going from 0 to y= (12 2x)/3= 4 (2/3)x. Since x= 3s and y= 2t, s= x/3 goes from 0 to 2 and, for each s, t goes from 0 to (4 (2/3)x)/2= 2 (1/3)x= 2 (1/3)(3s)= 2 s. That is, integrate $\displaystyle \int_0^6 \int_0^{2 s} .... dtds$. 

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integration, parameterization, surface 
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