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September 22nd, 2018, 09:24 AM   #1
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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
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September 22nd, 2018, 04:01 PM   #2
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Do you have a question?
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September 24th, 2018, 06:16 AM   #3
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Quote:
Originally Posted by mathman View Post
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
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September 24th, 2018, 09:05 AM   #4
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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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