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Homework Help: Show that r^2 = x^2 + y^2

  1. Apr 23, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider z=f(x,y), where x=rcosθ and y=rsinθ
    (This is a multi part multi variable calculus assignment question). I've just derived dz/dr and now I'm asked... to show that [itex]r^{2}=x^{2}+y^{2},

    2. Relevant equations

    3. The attempt at a solution

    I've drawn up a triangle with r as the hypotenuse, x on the x axis, y on the y axis.

    and then x=rcosθ=r * x/r = x by trig laws and then [itex]r^{2}=x^{2}+y^{2}[/itex] by pythagoras theorem etc

    this seems too easy. is there some other way to show this?
  2. jcsd
  3. Apr 24, 2013 #2


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    Not unless you want to prove the Pythagorean Theorem itself.
  4. Apr 24, 2013 #3


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    That doesn't have anything to do with ##z = f(x,y)##. It's just standard polar coordinate equations.$$
    x^2 + y^2 = r^2\cos^2\theta + r^2\sin^2\theta = r^2$$ $$
    \frac y x = \frac{r\sin\theta}{r\cos\theta}=\tan\theta$$
  5. Apr 24, 2013 #4
    yeah that's why I was confused. I don't get how I'm showing that by just plugging it into equations and referring to the polar equations.
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