 My Math Forum Line, Surface, and Volume elements- Parameters

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 September 22nd, 2018, 09:24 AM #1 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Line, Surface, and Volume elements- Parameters Line, Surface, and Volume elements. Let r be a vector in space r=(x,y,z) LINE x,y,z given in terms of parameter u. dr=r'du. dL=|r'|du SURFACE x,y,z given in terms of parameters u,v. In any coordinate direction, holding the other fixed, dr=r$\displaystyle _{u}$du, dr=r$\displaystyle _{v}$dv. dA=|r$\displaystyle _{u}$Xr$\displaystyle _{v}$|dudv VOLUME x,y,z given in terms of parametes u,v.w In any coordinate direction, holding the others fixed, dr=r$\displaystyle _{u}$du, dr=r$\displaystyle _{v}$dv, dr=r$\displaystyle _{w}$dw dV=|r$\displaystyle _{u}$Xr$\displaystyle _{v}$.r$\displaystyle _{w}$|dudvdw September 22nd, 2018, 04:01 PM #2 Global Moderator   Joined: May 2007 Posts: 6,850 Thanks: 742 Do you have a question? September 24th, 2018, 06:16 AM   #3
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Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 126

Quote:
 Originally Posted by mathman Do you have a question?
Yes, my own. Where do the L,A,V elements come from and how are they interrelated through parameters (coordinate transformations)? I answer it as a public service.

If you are happy with dxdydz=Jdudvdw, I would ignore OP.

J: Jacobian September 24th, 2018, 09:05 AM #4 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126 Nice example of Area element on a surface: Consider the parametrization x=u, y=v, z=z(u,v) Leave in this form to make it easier to follow the paradigm: $\displaystyle r-(u,v,z(u,v))$ $\displaystyle dr=r_{u}du= (1,0,z_{u})du$, $\displaystyle dr=r_{v}dv= (0,1,z_{v})dv$ $\displaystyle dA=|r_{u}Xr_{v}|dudv$ $\displaystyle dA=(1+z_{u}^{2}+z_{v}^{2})du$ Substituting x for u and y for v to get standard notation for this parametrization: $\displaystyle dA=(1+z_{x}^{2}+z_{y}^{2})dxdy$ Look it up. Tags elements, line, parameters, surface, volume Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post ricsi046 Real Analysis 3 April 28th, 2014 02:47 PM MarkFL Computer Science 9 April 3rd, 2014 05:33 PM Jhenrique Calculus 2 January 1st, 2014 05:08 PM winner Calculus 3 January 14th, 2011 12:30 PM winner Algebra 2 December 31st, 1969 04:00 PM

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