Conservative force in spherical coordinates

[Felipe Lincoln]

Homework Statement

Is $F=(F_r, F_\theta, F_\varphi)$ a conservative force?
$F_r=ar\sin\theta\sin\varphi$
$F_\theta=ar\cos\theta\sin\varphi$
$F_\varphi=ar\cos\varphi$

Homework Equations

$\nabla\times F=0$

The Attempt at a Solution

In this case we have to use the curl for spherical coordinates, but since it's clear that every component of this force can be described by cartesian coordinates I'm wondering if there's a way of doing the curl of cartesian coordinates, can we do this conversion ?

 

[kuruman]

What you have listed is not the spherical components of the vector, but the representation of the cartesian components using the spherical angles. For example $F_x=ar \sin\theta \cos \phi$, etc. So to find the curl, just take the appropriate derivatives and be sure to apply the chain rule of differentiation.

On edit: It looks like you have swapped the symbols and used $\varphi$ for the polar angle and $\theta$ for the azimuthal angle.

 

[Felipe Lincoln]

Right!
But why should I use the chain rule? I can just do some substitution like $x=r\sin\varphi\cos\theta$
And yes, I swapped the symbols, I'm used to do this way.
Thank you for the answer

 

[Felipe Lincoln]

Wait, I din't understand why you can say that $F_x=ar\sin\varphi\cos\theta$ (please use the symbols I introduced, it's my book convension). What the problem gave me ins't the radial componente of the force, in the case of $F_r$?

 

[kuruman]

What I'm saying is that the same vector $\vec F$ is written in spherical coordinates as
$\vec F = F_r \hat r~+~F_{\varphi}\hat {\varphi}~+~F_{\theta}\hat {\theta}$
and in Cartesian coordinates as
$\vec F = F_x ~\hat x~+~F_{y}~\hat {y}~+~F_{z}~\hat {z}$
the last expression can be written in terms of the spherical angles as
$\vec F = F \sin\varphi \cos \theta~\hat x~+~F \sin\varphi \sin\theta~\hat {y}~+~F \cos\varphi~\hat {z}$
where $F=\sqrt{F_x^2+F_y^2+F_z^2}=ar$.
I assumed you wanted to calculate the curl in the Cartesian representation, in which case you would have to find terms like $\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}$ which is where the chain rule comes in if you use the expressions of $F_x$ and $F_y$ that involve the spherical angles.

Note: Your book's convention is unconventional. Bear that in might if you use other books in the future.

 

[Felipe Lincoln]

I got it.
Thank you very much !