# Answer to Question #2185 in Vector Calculus for biju

Question #2185

Given the coordinate transformation

u

2u

u

Determine if the coordinate system is orthogonal.

u

_{1}= xy2u

_{2}= x^{2}+ y^{2}u

_{3}= 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

(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

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 = 1So we can say that given coordinate system isn't ortogonal

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