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.