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Could somebody explain this problem and solution(about partial derivatives)

  1. Apr 17, 2007 #1
    1. The problem statement, all variables and given/known data

    there is a question and a solution in this page;
    http://www.fen.bilkent.edu.tr/~otekman/math102/s03/m2q5.html

    Please firstly examine the question and solution. (5b)...
    There it says f(x,y)=z and x=g(r,teta) and y=h(r,teta) and asks fxx.
    He solves this problem by starting so;
    fx=zr*rx+zteta*tetax ...
    My question is that by writing fx such as above aren't you assume that r and teta are a function of x? However in the question it says x=f(r,teta); this means x is a function of r and teta then how can you write fx=zr*rx+zteta*tetax? Isn't there a contradiction or am i understanding something false?:confused:
     
  2. jcsd
  3. Apr 17, 2007 #2

    HallsofIvy

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    You are told that [itex]x= r cos(\theta)[/itex] so obviously x is a function of r and [itex]\theta[/itex] and therefore r and [itex]\theta[/itex] are functions of x. (It does not however anywhere say x= f(r,[/itex]\theta[/itex]).) As long as a function is invertible, you can work either way: If x= f(r), and r is invertible, then r= f-1(x) and dr/dx= 1/(dx/dr).\

    Here, you really have "polar coordinates": [itex]x= r cos(\theta)[/itex] and [itex]y= r sin(\theta)[/itex] so that [itex]r= \sqrt{x^2+ y^2}[/itex] and [itex]\theta= arctan(\frac{y}{x})[/itex]. You can do it either way.
     
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