Question

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.

Answer 1

The coordinate system is ortogonal if the metric tensor is diagonal.
Orthogonal coordinates never have off-diagonal terms in their metric tensor.
metric tensor G_ij=Summ(L=1 to N) (duL/dx_i+duL/dx_j)
(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