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hi, I have seen that christoffel symbol definition or logic is shown in different ways. For instance, in first attachment ( RED box) you can see a normal vector (n) next to the christoffel symbol, but in second image everything is same except that there is a normal vector. Is there a confusion? Which is the right representation of christoffel symbol? Why does these 2 different sources put a normal vector or not put a normal vector?

By the way first image is quoted from "wikipedia", other one is quoted from "Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces".

By the way first image is quoted from "wikipedia", other one is quoted from "Pavel Grinfeld Introduction to Tensor Analysis and the Calculus of Moving Surfaces".