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THE CARTAN STRUCTURAL EQUATIONS GREG W. ANDERSON
 

Summary: THE CARTAN STRUCTURAL EQUATIONS
GREG W. ANDERSON
The discussion takes place in the world of sophomore three-dimensional calculus
(div, grad, curl and all that).
Depending on context we let x1, x2, x3 or x, y, z denote the standard coordinates
in three-dimensional space.
Let
F = (F1, F2, F3), G = (G1, G2, G3), H = (H1, H2, H3)
be a right-handed frame of vector fields defined in some region of three-dimensional
Euclidean space. Recall that to be a right-handed frame the vector fields have to
satisfy
F G = F H = G H = 0,(1)
F F = G G = H H = 1,(2)
F G = H, G H = F, H F = G.(3)
For any vector field X we have the expansion
(4) X = (F X)F + (G X)G + (H X)H.
This will be a useful observation below.
Make some new vector fields
u = (u1, u2, u3), v = (v1, v2, v3), w = (w1, w2, w3)
by these rules:

  

Source: Anderson, Greg W. - School of Mathematics, University of Minnesota

 

Collections: Mathematics