Resolve the cartesian unit vectors into their cylindrical components

[JasonPhysicist]

Homework Statement

The problem is :''Resolve the cartesian unit vectors into their cylindrical components(using scale factors)

The Attempt at a Solution

It's simple to do the inverse(resolving cylindricl unit vectors into cartesian components),but I'm having some ''trouble'' with the above problem.
I know the answer is:

$$x=\rho\cos\varphi - \varphi\sin\varphi$$
$$y=\varphi\sin\varphi + \varphi\cos\varphi$$
$$z=z$$


Could someone shed some light?Thank you.





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[EngageEngage]

I think you're forgeting the vectors in the answer that you know. If you're converting the i,j,k (or x,y,z) unit vectors into their equivalent vectors in cylinderical coordinates, your answer will be in terms of 3 vectors, $$\vec{e_{r}},\vec{e_{\theta}},\vec{e_{z}}$$ (they might be presented differently in different texts though.)

The way to get to these base vectors is to look at the normals to the surfaces that are created when theta, r, and z are fixed constant. For example if you wish to find $$\vec{e_{r}}$$
you need to find normals to the surfaces what are created when r is constant -- that is cylindrical shells. You can do this by observation, or by simply noting the fact that a cylinder is:
$$x^{2}+y^{2} = r^{2}$$
and then simply finding a normal to it:
$$\vec{e_{r}} = cos(\theta)\vec{i}+sin(\theta)\vec{j}$$
you can do this for all of the vectors, and then solve the system of 3 equations to find the 3 vectors you are looking for