= Line element on a sphere
{title2=$ds^2=R^2(d\theta^2+\sin^2\theta\,d\phi^2)$}
In <spherical coordinates> on a <sphere> of radius $R$, the position is $X=R(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. Its coordinate tangent <vectors> satisfy $X_\theta\cdot X_\phi=0$, $|X_\theta|^2=R^2$, and $|X_\phi|^2=R^2\sin^2\theta$. 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 $R\sqrt{\dot\theta^2+\sin^2\theta\,\dot\phi^2}$. The vanishing azimuthal coefficient at a pole is a <coordinate singularity>, not a singularity of the <sphere>.
Back to article page
