Reference: Hobson, M.P., Efstathiou, G. P. & Lasenby, A. N. (2006), General Relativity: An Introduction for Physicists, Cambridge University Press. Problem 2.5. As another example of finding the metric tensor in a new coordinate system, we’ll look at the Mercator projection of the Earth’s surface onto a 2-d planar map. The Mercator projection is the one in which lines of longitude are mapped into straight vertical lines, perpendicular to the straight horizontal lines of latitude. It is also hideously inaccurate in terms of relative sizes of land masses, as it makes Greenland look larger than South America (the actual areas are for Greenland and for South America). To use conventional latitude and longitude, we note that longitude is equivalent to the spherical azimuthal angle , with corresponding to the Greenwich meridian. Latitude is (in radians) , where is the spherical polar angle. Therefore the line element in Earth coordinates can be obtained from the spherical line element: If is the horizontal coordinate on a Mercator map and is the vertical coordinate, the transformation is (as given in Hobson’s question): We need the derivatives with respect to and of and , which we can get from implicit differentiation. The space is approximately Euclidean near , which occurs when , that is, at the equator. The Mercator projection is essentially a projection onto a cylinder wrapped around the Earth, which touches the Earth at the equator, so this makes sense. Source.