Proving covariant component is physical component times scaling factor

[Monsterman222]

Homework Statement

The problem is from Mathematical Methods in the Physical Sciences, 3rd Ed. Ch10, Sec. 10, Q4. My question is a bit subtle as I have actually figured out the problem, just that I don't understand my solution. The problem reads:

4) What are the physical components of the gradient in polar coordinates? [See (9.1)]. The partial derivatives in (10.5) are the covariant components of \nabla u. What relation do you deduce between physical and covariant components? Answer the same questions for spherical coordinates, and for an orthogonal coordinate system with scale factors h_{1}, h_{2}, h_{3}.

Homework Equations

And we are given:

(9.1) \nabla u= \hat{e}_{r}\frac{\partial u}{\partial r} + \hat{e}_{\theta}\frac{1}{r}\frac{\partial u}{\partial \theta} + \hat{e}_{z}\frac{\partial u}{\partial z}

(10.5) \frac{\partial u}{\partial x'_{i}} = \frac{\partial u}{\partial x_{j}}\frac{\partial x_{j}}{\partial x'_{i}} = \frac{\partial x_{j}}{\partial x'_{i}}\frac{\partial u}{\partial x_{j}}

as well as the definition given in the book of a covariant vector:

(*) V'_{i} = \frac{\partial x_{j}}{\partial x'_{i}} V_{j}.

and

(**) \nabla u = \sum_{i=1}^{3} \hat{e}_{i} \frac{1}{h_{i}} \frac{\partial u}{\partial x_{i}}.

The Attempt at a Solution

I found that the relationship in these situations is that the covariant components of \nabla u are the physical components multiplied by the scaling factors. This is seen by looking at Eq. (**), for \hat{e}_{i}, the physical component is \frac{1}{h_{i}} \frac{\partial u}{\partial x_{i}} and the covariant component is \frac{\partial u}{\partial x_{i}}.

My question is: Does this relationship hold for any vector and not just gradients of scalar fields? How do we know? For example, I could define a vector V to be

\vec V = \hat{e}_{r}\frac{\partial u}{\partial r} + \hat{e}_{\theta}\frac{\partial u}{\partial \theta} + \hat{e}_{\phi}\frac{1}{r sin(\theta)}\frac{\partial u}{\partial \phi}.

Notice the middle term on right hand side has no scaling factor (this is not a gradient, I just made it up). This vector seems not to obey the relation between the physical components and covariant components.


Thanks!

 

[NewtonianAlch]

Sounds similar to me, I never understand my solutions; monkey see, monkey do.