 Line element on a sphere

In spherical coordinates on a sphere of radius $R$, the position is $X=R(\sin \theta \cos \varphi ,\sin \theta \sin \varphi ,\cos \theta )$. Its coordinate tangent vectors satisfy $X_{\theta }\cdot X_{\varphi }=0$, $|X_{\theta }{|}^2=R^2$, and $|X_{\varphi }{|}^2=R^2{\sin }^2\theta$. Expanding $|dX{|}^2$ proves the displayed line element, the first fundamental form of the sphere. Along a parametrized path it gives the arc length by integrating $R\sqrt{{\dot{\theta }}^2+{\sin }^2\theta {\dot{\varphi }}^2}$. The vanishing azimuthal coefficient at a pole is a coordinate singularity, not a singularity of the sphere.