#### martinbn

Science Advisor

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Probably not, unless you do something clever with limits. But that's beside the point, there will be questions, for which you need coordinates, that's for sure. But my wondering came from your post, where you said that for defining a manifold somewhere along the definition and axioms you will need them. So, is that the case? With charts and atlases you can get away without coordinates, but will they be needed at some point (before you want to do a calculation that requires them).Let us be less abstract and try to do something more concrete in differential geometry. Consider a two-dimensional sphere with unit radius, immersed in the 3-dimensional space with Euclid metric. Can you prove that the area of the sphere is ##4\pi##without using any coordinates? (I can't.)