Answer to Question #2185 in Vector Calculus for biju
Given the coordinate transformation
u[sub]1[/sub] = xy
2u[sub]2[/sub] = x[sup]2[/sup] + y[sup]2[/sup]
u[sub]3[/sub] = z,
Determine if the coordinate system is orthogonal.
The coordinate system is ortogonal if the metric tensor is diagonal. Orthogonal coordinates never have off-diagonal terms in their metric tensor. metric tensor Gij=Summ(L=1 to N) (duL/dxi+duL/dxj) (x y z)=(x1 x2 x3) So we must check if the tensor is diagonal G11 = y2+1+0 = 1+y2 G12 = xy+xy+0 = 2xy = G21 G13 = 0+0+0=0 = G31 G22 = 0+y2+0 = y2 G33 = 0+0+1 = 1 So we can say that given coordinate system isn't ortogonal