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Post by Admin on Dec 22, 2017 6:49:18 GMT
To me it doesn't make sense to say, as Needham does in this exercise, "...show that the slope of this steepest tangent plane is..."
All the tangent lines through a given point on a modular surface lie in the same tangent plane and we want to find the slope of the steepest tangent line at the given point.
So I think part (ii) should be rewritten replacing the word "plane" in the second sentence of part (ii) by "line".
Vasco/Admin
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Gary
GaryVasco
Posts: 3,352 
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Post by Gary on Dec 31, 2017 1:04:41 GMT
To me it doesn't make sense to say, as Needham does in this exercise, "...show that the slope of this steepest tangent plane is..." All the tangent lines through a given point on a modular surface lie in the same tangent plane and we want to find the slope of the steepest tangent line at the given point. So I think part (ii) should be rewritten replacing the word "plane" in the second sentence of part (ii) by "line". Vasco/Admin Vasco, In my answer, I included the following comments: This question strikes me as a bit vague and hard to parse because it shifts from “tangent line” to “tangent plane”. Of course, the steepest tangent line to the modular surface lies in a plane tangent to the modular surface at the same point, and both line and plane have the same slope. I think by “steepest slope”, Needham does not mean the steepest slope anywhere in the modular surface. I think he means that of all the tangent lines one could plot from a point directly above p on the modular surface (and all lying in the tangent plane), the line that aligns with the direction of p is the steepest. Gary |
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Post by Admin on Dec 31, 2017 6:34:01 GMT
To me it doesn't make sense to say, as Needham does in this exercise, "...show that the slope of this steepest tangent plane is..." All the tangent lines through a given point on a modular surface lie in the same tangent plane and we want to find the slope of the steepest tangent line at the given point. So I think part (ii) should be rewritten replacing the word "plane" in the second sentence of part (ii) by "line". Vasco/Admin Vasco, In my answer, I included the following comments: This question strikes me as a bit vague and hard to parse because it shifts from “tangent line” to “tangent plane”. Of course, the steepest tangent line to the modular surface lies in a plane tangent to the modular surface at the same point, and both line and plane have the same slope. I think by “steepest slope”, Needham does not mean the steepest slope anywhere in the modular surface. I think he means that of all the tangent lines one could plot from a point directly above p on the modular surface (and all lying in the tangent plane), the line that aligns with the direction of p is the steepest. Gary Gary I agree entirely with your interpretation, that's what I was trying to say in my post. Thanks. I am currently writing up this exercise and I hope to complete it within the next few weeks and then, having started reading chapter 10, continue with that. Vasco |
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