Question #2185

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.

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 G_{ij}=Summ(L=1 to N) (duL/dx_{i}+duL/dx_{j})

(x y z)=(x_{1} x_{2} x_{3})

So we must check if the tensor is diagonal

G_{11} = y^{2}+1+0 = 1+y^{2}

G_{12} = xy+xy+0 = 2xy = G_{21}

G_{13} = 0+0+0=0 = G_{31}

G_{22} = 0+y^{2}+0 = y^{2}

G_{33} = 0+0+1 = 1

So we can say that given coordinate system isn't ortogonal

Orthogonal coordinates never have off-diagonal terms in their metric tensor.

metric tensor G

(x y z)=(x

So we must check if the tensor is diagonal

G

G

G

G

G

So we can say that given coordinate system isn't ortogonal

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