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.
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
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: