[FEA Newbie] "Jacobian matrix" when topological dim different from embedding dim

Hi all,
I experiencing some difficulties while learning the finite element
method and its implementation: how to deal with the case where the
topological dimension of the reference element (e.g. 1 for segment, 2
for triangle and 3 for tetrahedron) is different from the embedding
space dimension?
(i) Let us consider a phi function to be integrated over triangles lying
in 2D space. x and y are embedding coordinates. xi and eta are reference
element coordinates.
dphi/dx = dphi/dxi * dxi/dx + dphi/deta * deta/dx
dphi/dy = dphi/dxi * dxi/dy + dphi/deta * deta/dy
J is the jacobian matrix (J^-1 --> dxi/d[x,y] and deta/d[x,y])
dxdy = dej(J) * dxideta
(ii) Let us consider a phi function to be integrated over triangles
lying in 3D space. x, y and z are embedding coordinates. xi and eta are
reference element coordinates.
dphi/dx = dphi/dxi * dxi/dx + dphi/deta * deta/dx
dphi/dy = dphi/dxi * dxi/dy + dphi/deta * deta/dy
dphi/dz = dphi/dxi * dxi/dz + dphi/deta * deta/dz
dxdydz = ?
So main problem is to obtain dxi/d[x,y,z] and deta/d[x,y,z]. From an
implementation point of view the case (i) is just a matter of inverting
a matrix. What about this case ?
I am a little bit lost and I would like some pointers (www, books or
inline explanation).
Thanks all & sincerely.
PM
P.S. : I have a rather weak physics and mathematics background so feel
free to start from basics again.
P.S. : Also posted on sci.math.num-analysis & sci.engr.mech
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PM
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