I'm going through Introduction_to_Tensor_Calculus by Wiskundige_Ingenieurstechnieken.(adsbygoogle = window.adsbygoogle || []).push({});

I want to find the covariant and contravariant components of the cylindrical cooridnates.

Ingerniurstechnieken sets tangent vectors as basis (E_{1}, E_{2}, E_{3}). He also sets the normal vectors as another basis (E^{1}, E^{2}, E^{3})

The coorindates relationships are:

x=r*cosθ

y=r*sinθ

z=z

r=sqrt(x^{2}+y^{2})

θ=arctan(y/x)

z=z

I'm setting R= vector r

R=[r*cosθ, r*sinθ, 1]

I found:

E_{1}= [cosθ, sinθ, 0]

E_{2}= [-r*sinθ, r*cosθ, 0]

E_{3}= [0, 0, 1]

He says to take gradient of r, θ, and z for normal basis (E^{1}, E^{2}, E^{3}). What do I take the gradient of to get this basis?

Is it ∇r=(1, 0, 0)

∇θ=(0, 1, 0)

∇z=(0, 0, 1)?

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# I Finding Covariant and Contravariant components of Cylin Coor

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