Euclidean Sub-manifolds - Certain subsets of Euclidean space admit a suitable theory of calculus. The class of subsets we consider are the so-called sub-manifolds. Linear subspaces are a particular example, but not all (in fact most) sub-manifolds are not linear subspaces.

Inverse Function Theorem - The Inverse Function Theorem is a cornerstone theorem in nonlinear analysis. It is essential for the development of sub-manifolds. There are several equivalent formulations such as the Implicit Function Theorem, the Immersion Theorem and the Submersion theorem. We will discuss each of these and see that they are in fact equivalent to each other. We won't focus too much on the proof itself however as this is result best dealt with in an analysis course.

Embedding of Sub-manifolds - Let us now investigate the manner in which an embedded sub-manifold is embedded in Euclidean space. There is a compatibility between the intrinsic differentiable structure from local parametrisations and the ambient differential structure from the surrounding Euclidean space. We will also see how the regularity condition of injective differential is precisely what is required to allow us to develop a theory of calculus on an embedded sub-manifold. The main tool used is the *Inverse Function Theorem* and several variants such as the *Implicit Function Theorem*.