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?

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# Puzzling paradox

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