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Cylindrical coordinates:

Cylindrical coordinates are essentially the usual polar coordinates in the plane, but with vertical z added in. The conversions are familiar

x = rcos(ϕ), y = rsin(ϕ), z = z

r^2 = x^2 + y^2, tan(ϕ) = y/z, z = z

Sometimes rho is used rather than r. Adjust your results to read s, (as in your diagram), instead.

Cylindrical unit vectors

Composed of perpendicular components in directions i, j or k

The sum of components making r(hat) can be seen from the diagram

r(hat) = cos(ϕ) i + sin(ϕ) j

ϕ(hat) is perpendicuar to that and given by

∂/∂ϕ[r(hat)] and so we have

ϕ(hat) = -sin(ϕ) i + cos(ϕ) j

The last one is easy

z(hat) = z(hat)

Remember to use s instead of r in your version.

Here is a video explanation of the same thing

https://www.youtube.com/watch?v=BuKR7Y2s87Q

Well, let me go back to school and take the course, take notes, and pay attention....

Well,

x = r cos(φ)

y = r sin(φ)

z = z

hope it' ll help !!