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Post by Admin on Mar 7, 2016 17:45:25 GMT
I have published exercise 17 of chapter 5, but only parts (i) to (v). I hope to publish part (vi) at a later date. I have also posted some further thoughts on this exercise in the errata section.
Vasco
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Mar 8, 2016 23:23:55 GMT
I have published exercise 17 of chapter 5, but only parts (i) to (v). I hope to publish part (vi) at a later date. I have also posted some further thoughts on this exercise in the errata section. Vasco Vasco, I have read your answer to 5:17 very closely. It all looks correct and it seems to me to answer the question properly. It appears that the key for all three functions $e^z, log(z), and z^m$ is the following: For the curvature of f(s) ($\overline{\kappa}$) to vanish at p, f(S) must be a line segment. The solution therefore requires finding the constraints under which f(S) is a line segment. For $e^z$, this is true whenever S is parallel to the real line. For log(z) and $z^m$, it is true whenever S is parallel to the ray from 0 to p. I could not solve it because it never dawned on me that f(S) had to be a line segment, even though it is blindingly obvious that only a straight line has zero curvature. In case there are others who might think as perversely as I do, I think I let myself in for confusion by misreading paragraph 2, p. 238: “The first term says that the mapping will introduce curvature even where none is originally present.” I didn’t allow for the fact that it might introduce zero curvature. I thought perhaps I had gotten into some strange new mathematical territory and I should suspend judgement for a while. I agree with your comments on "without calculations" in the errata. Obviously there has to be a bit of algebra to decide that S lies on the real line or on a ray through p. Gary |
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Post by Admin on Mar 9, 2016 18:02:19 GMT
I have published exercise 17 of chapter 5, but only parts (i) to (v). I hope to publish part (vi) at a later date. I have also posted some further thoughts on this exercise in the errata section. Vasco Vasco, I have read your answer to 5:17 very closely. It all looks correct and it seems to me to answer the question properly. It appears that the key for all three functions $e^z, log(z), and z^m$ is the following: For the curvature of f(s) ($\overline{\kappa}$) to vanish at p, f(S) must be a line segment. The solution therefore requires finding the constraints under which f(S) is a line segment. For $e^z$, this is true whenever S is parallel to the real line. For log(z) and $z^m$, it is true whenever S is parallel to the ray from 0 to p. I could not solve it because it never dawned on me that f(S) had to be a line segment, even though it is blindingly obvious that only a straight line has zero curvature. In case there are others who might think as perversely as I do, I think I let myself in for confusion by misreading paragraph 2, p. 238: “The first term says that the mapping will introduce curvature even where none is originally present.” I didn’t allow for the fact that it might introduce zero curvature. I thought perhaps I had gotten into some strange new mathematical territory and I should suspend judgement for a while. I agree with your comments on "without calculations" in the errata. Obviously there has to be a bit of algebra to decide that S lies on the real line or on a ray through p. Gary Gary Thanks for the detailed feedback. Much appreciated. Vasco |
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