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I follow the book Lectures on Differential Geometry by Chern, do not understand the proof.

The proof is

the Jacobian Matrix of the transformation between coordinates of two charts has positive determinant (oriented charts), so the smaller Jacobian Matrix with one row and one column deleted (corresponding to the only one coordinate axis that runs away from the boundary) has positive determinant.

Please do me a favour by explaining the proof clearly, or give me another easier proof.