An online book on differential geometry
which I like better than the Do Carmo textbook.
In this book there is a careful statement of the Inverse and Implicit
Function Theorems on page 3 and a proof that the
three definitions of a regular surface are equivalent on page 6.
lecture notes on manifolds and maps
that I used whenever I taught Math 761.
They are too advanced for 561 but they do contain complete proofs
of the Inverse and Implicit Function Theorems and
the Existence and Uniqueness Theorem for Ordinary Differential Equations.