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Answer to Question #2185 in Vector Calculus for biju

Question #2185
Given the coordinate transformation
u1 = xy
2u2 = x2 + y2
u3 = z,
Determine if the coordinate system is orthogonal.
Expert's answer
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

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