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
Banned Camp

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.

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