I want to integrate around a closed circular path on xy plane around the origin. Say the radius is b. So(adsbygoogle = window.adsbygoogle || []).push({});

##\oint d\vec l## where ##d\vec l=\hat{\phi}b d\phi##

1) If I just use polar( or spherical or even cylindrical) coordinates. R=b and

[tex]\oint d\vec l\;=\;\hat{\phi}\int_0^{2\pi} b d\phi\;=\;2\pi b[/tex]

2) Using rectangular co where ##\hat{\phi}=-\hat x \sin \phi +\hat y \cos \phi##

[tex]\oint d\vec l\;=\;b\int_0^{2\pi} (-\hat x \sin\phi+\hat y \cos \phi) d\phi\;=\;0[/tex]

What did I do wrong?

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# Confused about line integral.

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