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Question about parameterizing curve of intersection. 
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#1
Jun1012, 12:32 PM

P: 122

I couldn't find any resources in my book or online dedicated to this subject. I honestly don't even know where to begin for this problem.
1. The problem statement, all variables and given/known data Let [itex] f(x,y) = 4 / (1+ x^2 + y^2) [/itex] and let S be the surface given by the graph of f(x,y) b) Let C2 denote the curve in the xyplane given by [itex] r(t)= t, 3/2 − t^2[/itex] and let C denote the curve on the surface S which has C2 as its shadow in the xyplane. Find the parametric equations r = r(t) for C 2. Relevant equations 3. The attempt at a solution 


#2
Jun1012, 01:28 PM

P: 428

how about the "graph" over the curve,
[tex](r(t),f(r(t))=(x(t),y(t),z(x(t),y(t)))[/tex] Then it is a curve, on the surface, and it's shadow is r(t), correct? 


#3
Jun1012, 01:53 PM

P: 122

Hrmm.. I'm not too sure honestly. I think this all has to do with arclength/curvature. If that is correct then I think I will go study more about that and see if I can't figure it out



#4
Jun1012, 02:18 PM

P: 428

Question about parameterizing curve of intersection.
I would not discourage you from exploring the concepts, and so develop your understanding of the subject as a whole. However, I think you'll find, while my hint is somewhat abstract, it is more or less correct, and that curvature and arclength do not apply here. But again, please do investigate and compare the concepts!
Also, my use of the word graph is not a bad definition for you to understand, Stewart uses it in his textbook on calculus; here is the definition of graph (there are other definitions) on wikipeida: http://en.wikipedia.org/wiki/Graph_of_a_function In other words, what is the graph associated with the function (x,y) > z=f(x,y) 


#5
Jun1012, 02:56 PM

P: 122

I dont deal well with this sort of abstraction.
In my mind what you are saying is we can come up with a new function which is simply our old function r(t) plus a new variable which is the sum of the variables of our original function. Correct me if I am wrong. But it would appear to me that we need our original function to come up with the parameters, since S is the measure of the surface of [itex] f(x,y)=4/(1+x2+y2) [/itex]. Oh and one last thing. A "shadow" is simply a projection correct? 


#6
Jun1012, 03:53 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,348

There is no abstraction here! You are given x and y in terms of t and told how to calculate z in terms of x and y. So what is z in terms of t? It is just basic algebra.



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