A simple (?) vector problem

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In summary, the textbook and the Attempt at a Solution both agree that B should be 1/r\left(\frac{\partial (A_{\theta}r)}{\partial r}-\frac{\partial A_r}{\partial \theta}\right)\hat{\phi}.
  • #1
_Andreas
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Homework Statement



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

[tex]A_r[/tex] and [tex]A_{\theta}[/tex] are the components of the basis vectors [tex]\hat{r}[/tex] and [tex]\hat{\theta}[/tex].

The Attempt at a Solution



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

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

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

[tex]\frac{\partial (A_{\theta}r)}{\partial r}[/tex]

part. Why isn't it

[tex]r\frac{\partial (A_{\theta})}{\partial r}[/tex]?
 
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  • #2
I agree with you and I don't agree with the textbook.
 
  • #3
Dick said:
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.
 
  • #4
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.
 
  • #5
Dick said:
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 [tex]r[/tex] and [tex]\theta[/tex]).
 
  • #6
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
 
  • #7
Dick said:
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 [tex]r[/tex] in front of [tex]\bm{E}_\theta[/tex] in the determinant (see the link I provided) comes from, though.
 
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1. What is a vector?

A vector is a mathematical quantity that has both magnitude (size) and direction. In physics, it is often used to represent physical quantities such as velocity, force, and displacement.

2. How do you solve a vector problem?

To solve a vector problem, you first need to identify the given information and what is being asked. Then, you can use vector addition, subtraction, or scalar multiplication to manipulate the vectors and find the desired result.

3. What is the difference between a scalar and a vector?

A scalar is a quantity that has only magnitude (size) and no direction, while a vector has both magnitude and direction. Examples of scalars include time and temperature, while examples of vectors include displacement and velocity.

4. Can vectors be negative?

Yes, vectors can be negative. A negative vector indicates that the magnitude and direction of the vector are in the opposite direction from the positive vector.

5. What are the different types of vector problems?

There are three main types of vector problems: 1) adding or subtracting vectors, 2) finding the components of a vector, and 3) finding the magnitude and direction of a vector. Each type of problem requires a different approach to solve.

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