Legality and Generality of a Simplifying Method

In summary, the conversation discusses a method for simplifying a vector expression involving a spherical unit vector and two vectors, and the question of its legality and generality. One person proposes a different approach where the direction of one vector is fixed while the other is freely chosen based on the spherical symmetry of the integration volume. Another person suggests splitting one vector into parallel and orthogonal components and adding them together, which results in a final answer that depends only on the cross product of the two vectors. This method is found to be successful in simplifying the expression.
  • #1
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I have a question regarding the legality and generality of the following method for simplifying a vector expression. The specific expression that arose my question is ##\left[\left(\vec{P}\cdot\hat{r}\right)\left(\hat{r}\times\vec{M}\right) + \left(\vec{M}\cdot\hat{r}\right)\left(\vec{P}\times\hat{r}\right)\right]## where ##\hat{r}## is the spherical unit vector pointing away from the origin. This is part of a longer expression that needs to be integrated over a certain volume. I know that it can be simplified using a lengthy manipulation involving multiple usages of triple product formulas but even then, one still needs to assume that ##\vec{M}\times\vec{P}## lies along some known axis (like the z-axis) to do the integral and then bring it back into coordinate free form (if it is relevant or interesting to someone, this method can be found http://physicspages.com/2014/06/20/momentum-in-a-magnetized-and-polarized-sphere/). What I propose, however, is to suppose that, for instance, ##\vec{M} = M\hat{z}## and ##\vec{P} = P\hat{y}##. Then, plug these into the above expression, get the final answer (since the expression will be integrated over a volume, there will be no dependence on the spherical coordinates) and only then express the result in terms of ##\vec{M}\times\vec{P}##.
Is there any loss of generality in this approach? (It does give the correct answer, I'm worried about its legality and generality).

By the way, if there is any "test" I can perform to test for generality in such case, I would be glad to know about it.

Any comments/corrections/suggestions will be greatly appreciated!
 
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  • #2
You don't need assumptions about MxP, but you can freely choose the direction of MxP if the integration volume has a spherical symmetry.
I don't think you can fix the direction of both M and P independently. That is beyond the freedom the spherical symmetry gives. Imagine P=M, then the integral is zero.

You can split M into a part parallel to P and a part orthogonal to P, however, this is always possible. Both components are easier to evaluate (one is zero) and the final result is linear in M, so you can just add the two components. That should mean your result depends on MxP only, right.
 
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  • #3
mfb said:
You don't need assumptions about MxP, but you can freely choose the direction of MxP if the integration volume has a spherical symmetry.
I don't think you can fix the direction of both M and P independently. That is beyond the freedom the spherical symmetry gives. Imagine P=M, then the integral is zero.

You can split M into a part parallel to P and a part orthogonal to P, however, this is always possible. Both components are easier to evaluate (one is zero) and the final result is linear in M, so you can just add the two components. That should mean your result depends on MxP only, right.
Sorry for the late reply and thank you very much. I have finally got around to redoing this problem with your method and succeeded.
 

1. What is the legality of using a simplifying method in scientific research?

The legality of using a simplifying method in scientific research depends on the specific method being used and the ethical guidelines of the scientific community. It is important to carefully consider the potential impact of using a simplifying method and ensure that it does not compromise the accuracy or validity of the research.

2. How do scientists determine the generality of a simplifying method?

The generality of a simplifying method is typically determined through rigorous testing and comparison with other methods. Scientists often conduct experiments using the simplifying method and then compare the results with those obtained using more complex methods. If the results are similar, it can be concluded that the simplifying method has a high level of generality.

3. Is it ethical to use a simplifying method to save time in research?

The ethics of using a simplifying method in research depends on the specific circumstances and the potential impact on the validity and accuracy of the research. While time-saving may be a factor, it is important for scientists to prioritize the integrity of their research and carefully consider the potential consequences of using a simplifying method.

4. What are some potential drawbacks of using a simplifying method in scientific research?

Some potential drawbacks of using a simplifying method in scientific research include the potential for biased results, inaccurate conclusions, and limited applicability to real-world situations. Additionally, using a simplifying method may overlook important variables or nuances that could significantly impact the outcome of the research.

5. How can scientists ensure that a simplifying method does not compromise the validity of their research?

To ensure that a simplifying method does not compromise the validity of their research, scientists should carefully design experiments and compare the results with those obtained using more complex methods. They should also consider potential biases and limitations of the simplifying method and take steps to mitigate them. Additionally, it is important for scientists to clearly communicate the use of a simplifying method and its potential impact on the research findings.

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