Why Does the Curl Calculation in Cylindrical Coordinates Include a Factor of r?

[_Andreas]

Homework Statement

Determine the vector $$\bm{B}=\left(\frac{\partial A_{\theta}}{\partial r}-\frac{1}{r}\frac{\partial A_r}{\partial \theta}\right)\hat{\phi}$$

$$A_r$$ and $$A_{\theta}$$ are the components of the basis vectors $$\hat{r}$$ and $$\hat{\theta}$$.

The Attempt at a Solution

I just calculated the differentials in the expression for B above, but that gave me a factor $$1/r$$ too much in the answer. My textbook rewrites B as

$$\bm{B}=\frac{1}{r}\left(\frac{\partial (A_{\theta}r)}{\partial r}-\frac{\partial A_r}{\partial \theta}\right)\hat{\phi}$$.

They've broken out a factor $$1/r$$ before differentiating, but I don't understand the

$$\frac{\partial (A_{\theta}r)}{\partial r}$$

part. Why isn't it

$$r\frac{\partial (A_{\theta})}{\partial r}$$?

 

[Dick]

I agree with you and I don't agree with the textbook.

 

[_Andreas]

I agree with you and I don't agree with the textbook.

Unfortunately for us, I have another textbook that agrees with the first. I don't think it's likely that two professional physicists makes exactly the same mistake on such a (I guess) basic thing.

 

[Dick]

Well, if I put A_theta=1 and A_r=1, I certainly don't get the same result for B for the two expressions. That's about all I can say.

 

[_Andreas]

Well, if I put A_theta=1 and A_r=1, I certainly don't get the same result for B for the two expressions. That's about all I can say.

I know, it's a mystery. Perhaps I should provide some more information. B is the cross product of the operator del and another vector, A, expressed in polar coordinates (B is only a function of $$r$$ and $$\theta$$).

 

[Dick]

It sounds like you are computing the curl of A. If you are working in cylindrical coordinates, and A is independent of z, and the z component of A is independent of r and theta, then your second expression is the curl if you replace the theta hat with a z hat. That's a near as I can get to figuring out what you are up to. http://mathworld.wolfram.com/CylindricalCoordinates.html

 

[_Andreas]

It sounds like you are computing the curl of A. If you are working in cylindrical coordinates, and A is independent of z, and the z component of A is independent of r and theta, then your second expression is the curl if you replace the theta hat with a z hat. That's a near as I can get to figuring out what you are up to. http://mathworld.wolfram.com/CylindricalCoordinates.html

Yes, it's the curl I'm trying to calculate. I see now that I've been working with an incorrect http://hyperphysics.phy-astr.gsu.edu/Hbase/curl.html#c2". I still don't know where the factor $$r$$ in front of $$\bm{E}_\theta$$ in the determinant (see the link I provided) comes from, though.