Cartesian unit vectors in terms of cylindrical vectors

[Komekami]

How do I express ex,ey,ez in terms er,eθ,eZ?
r=(x^2+y^2)^1/2,θ=arctan(y/x),Z=z
A(r,θ,z)
∂A/∂x=x/(x^2+y^2)^1/2er+(-y)/(x^2+y^2)eθ=cosθer-(sinθ/r)eθ
ex=(∂A/∂x)/|∂A/∂x| I should get ex as cosθer-sinθeθ, but I don't get ex correctly.
am i doing this wrong?

 

[kuruman]

How do I express ex,ey,ez in terms er,eθ,eZ?

Make yourself a drawing with $\hat{e}_x$ horizontal and $\hat{e}_y$ vertical. In this drawing add $\hat{e}_r$ at angle $\theta$ w.r.t. $\hat{e}_x$. Add $\hat{e}_{\theta}$ perpendicular to $\hat{e}_r$. Study the drawing and find the projections of $\hat{e}_x$ on $\hat{e}_r$ and $\hat{e}_{\theta}$. You should get what you expect to get.