Hi I am having a bit of difficulty working with plane polar co-ordinates.(adsbygoogle = window.adsbygoogle || []).push({});

We have:

[tex] r^2 = x^2 + z^2 [/tex]

[tex] x = r cos \theta [/tex]

[tex] z=r sin \theta [/tex]

I wish to find [tex] \frac {\partial r} {\partial x} [/tex]

Using [tex] r^2 = x^2 + z^2 [/tex]

We have:

[tex] \frac {\partial (r^2)} {\partial x} = \frac {\partial (x^2)} {\partial x} + \frac {\partial (z^2)} {\partial x} [/tex]

Thus [tex] 2r\frac {\partial r} {\partial x} = 2x [/tex]

[tex] \frac {\partial r} {\partial x} = \frac {x} {r} = \frac {r cos \theta} {r} [/tex]

Therefore [tex]\frac {\partial r} {\partial x} = cos \theta [/tex]

But if we find [tex] \frac {\partial r} {\partial x} [/tex] using [tex] x = r cos \theta [/tex]

We have:

[tex] r = \frac {x} {cos \theta} [/tex]

Therefore [tex]\frac {\partial r} {\partial x} = \frac {1} {cos \theta} [/tex]

What is going on here? Which answer is wrong and why?

**Physics Forums - The Fusion of Science and Community**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Puzzling paradox

**Physics Forums - The Fusion of Science and Community**