I am a beginner student in Differential Geometry and reading the book Introduction to smooth manifolds by Jhon lee. I am trying to understand what does in mean to say something is coordinate independent. For example, when he talks about line integral as an application of covector fields he says "covectors make coordinate independent sense of line integrals". Does that simply mean in defining line integrals we do not use coordinate charts? or Does it have more meaning to it? Also while motivating alternating tensors he says they help in making coordinate independent interpretation of multiple integrals.
To say that something is coordinates independent means that it does not depend from the coordinates we chose to represent such thing.
the fact that two vectors in a given inner product space are orthogonasl, is coordinate independent;
the lenght of an arc ofa line is independent from the coordinate chosen to represents the line;
the Gaussian curvature of a surfaceat a point is independent from the chart chosen to represents the surface in a negboorod of the point.