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September 22nd, 2018, 09:24 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  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 
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September 24th, 2018, 06:16 AM  #3 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  
September 24th, 2018, 09:05 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
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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elements, line, parameters, surface, volume 
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