Post by Admin on Mar 9, 2016 15:25:40 GMT
Following the comments by Gary and looking again at exercise 20, it is clear that there is an inconsistency between the results of parts (i) and (iii) and part (iv). Parts (i) and (iii) allow $\widetilde{\kappa}$ to be positive or negative in accord with the treatment of curvature in section IX of the book on pages 234-241, whereas the result in part (iv) only allows $\widetilde{\kappa}$ to be positive.
I have edited my published solution to reflect this small error in the statement of the exercise.
The exercise could be amended in several ways to make it consistent. I have today looked at a derivation of the result in part (iv) of the exercise using vector methods, in the book "Elementary Vector Analysis" by C. E. Weatherburn. The author assumes at the beginning that the curvature is a positive number whereas in the exercise no such assumption is made. This is what leads to the inconsistency in the exercise, mentioned above.
Following the logic of the exercise as written we find that $\widetilde{\kappa}=\frac{\textbf{v}\times\dot{\textbf{v}}}{|\textbf{v}|^3}$, which allows positive and negative values for $\widetilde{\kappa}$. In my amended solution I have just pointed out that the result in part (iv) of the exercise is a solution with only positive values.
We could amend the exercise so that we start by insisting that $\widetilde{\kappa}$ be positive, but this would require us to change the results in parts (i) and (iii) to $\widetilde{\kappa}=\frac{|Im(\dot{v}/v)|}{|v|}$ and $\widetilde{\kappa}=\frac{|Im(\overline{v}\dot{v})|}{|v|^3}$, respectively.
Another alternative amendment would be to change the wording of part (iv) to say:
"Deduce that if we are only interested in the magnitude of the curvature then it may also be written vectorially as...".
I'm not sure which is best but the main point is to be aware of the inconsistency.
Vasco
I have edited my published solution to reflect this small error in the statement of the exercise.
The exercise could be amended in several ways to make it consistent. I have today looked at a derivation of the result in part (iv) of the exercise using vector methods, in the book "Elementary Vector Analysis" by C. E. Weatherburn. The author assumes at the beginning that the curvature is a positive number whereas in the exercise no such assumption is made. This is what leads to the inconsistency in the exercise, mentioned above.
Following the logic of the exercise as written we find that $\widetilde{\kappa}=\frac{\textbf{v}\times\dot{\textbf{v}}}{|\textbf{v}|^3}$, which allows positive and negative values for $\widetilde{\kappa}$. In my amended solution I have just pointed out that the result in part (iv) of the exercise is a solution with only positive values.
We could amend the exercise so that we start by insisting that $\widetilde{\kappa}$ be positive, but this would require us to change the results in parts (i) and (iii) to $\widetilde{\kappa}=\frac{|Im(\dot{v}/v)|}{|v|}$ and $\widetilde{\kappa}=\frac{|Im(\overline{v}\dot{v})|}{|v|^3}$, respectively.
Another alternative amendment would be to change the wording of part (iv) to say:
"Deduce that if we are only interested in the magnitude of the curvature then it may also be written vectorially as...".
I'm not sure which is best but the main point is to be aware of the inconsistency.
Vasco