How to write basis vectors in Cartesian coordinates in terms of cylindrical coordinates?

Asked Oct 18 '18 at 22:09

Active Oct 18 '18 at 22:09

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Given a Cartesian coordinate system with basis vectors (e_x, e_y, e_z)

and a Cylindrical coordinate system with basis vectors (e_r, e_θ, e_z)

Why and how does:

e_x = e_rCos(θ) - e_θSin(θ) ?

e_y = e_rSin(θ) + e_θCos(θ) ?

This is from a derivation of the del operator in cylindrical coordinates from my lecture notes.

I cannot for the life of me figure this out. Shouldn't e_x = e_rCos(θ) and e_y = e_rSin(θ)?

What is it with the extra Sin and Cos?

A simple, detailed derivation/ explanation would be very helpful.

asked Oct 18 '18 at 22:09

Usama Ahmed

This seems related https://math.stackexchange.com/questions/2947583/graphically-representing-vectors-with-polar-unit-vectors-without-converting-to-c – David K Oct 18 '18 at 22:19
The differential of a function $f(u,v)$ is $f_u du + f_v dv$. Apply this to $x=r\cos(\theta)$ to get $dx = \cos(\theta) dr - \sin(\theta) d\theta$, and similarly for $y = r\sin\theta$. – Nick Oct 18 '18 at 23:01

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