 
Summary: THE CARTAN STRUCTURAL EQUATIONS
GREG W. ANDERSON
The discussion takes place in the world of sophomore threedimensional calculus
(div, grad, curl and all that).
Depending on context we let x1, x2, x3 or x, y, z denote the standard coordinates
in threedimensional space.
Let
F = (F1, F2, F3), G = (G1, G2, G3), H = (H1, H2, H3)
be a righthanded frame of vector fields defined in some region of threedimensional
Euclidean space. Recall that to be a righthanded 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:
