Manifolds are sets which resemble ℝ^{n} locally. A manifold with boundary resembles either ℝ^{n} or half of ℝ^{n} locally. Here only the first concept will be discussed because to do the second satisfactorily, one needs the theorem on invariance of domain which has not been presented. There are many other ways to generalize what is presented here. For example, it will be assumed that the atlas is finite. It suffices to assume it is locally finite. It is not necessary to assume the manifold is a compact subset of ℝ^{m} as is done below and so forth. To make this concept of a manifold more precise, here is a definition.
Definition 10.2.1 Let Ω ⊆ ℝ^{m}. A set U, is open in Ω if it is the intersection of an open set from ℝ^{m} with Ω. Equivalently, a set U is open in Ω if for every point x ∈ U, there exists δ > 0 such that if
Here is a definition which will be used.
Definition 10.2.2 Let V ⊆ ℝ^{n}. C^{k}
Definition 10.2.3 A closed and bounded subset of ℝ^{m} Ω, will be called an n dimensional manifold, n ≥ 1, if there are finitely many sets U_{i}, open in Ω and continuous one to one on U_{i} functions R_{i} ∈ C^{0}
Example 10.2.4 A typical example is the boundary of a bounded open set. For example, you could consider S^{p−1}, the unit sphere which is the boundary of the unit ball in ℝ^{p}.
For example in case p = 3, you could have R_{1}

Then R_{1}^{−1}
We could include this line in other charts as follows.

Now these two charts,

Now these two include the North and South poles but miss the equator.
Thus an atlas for the sphere is as follows.
Next is the concept of an oriented C^{k} manifold, k ≥ 1. Orientation can be defined for general C^{0} manifolds using the topological degree, but the reason for considering this, at least here, is for the sake of its interaction with integrals on the manifold and so typically something more than C^{0} is desired.
Definition 10.2.5 An n dimensional C^{k} manifold Ω is called orientable if there exists an atlas,
 (10.1) 
The mappings, R_{i} ∘ R_{j}^{−1}are called the overlap maps. An atlas satisfying 10.1 is called an oriented atlas.
In calculus, you probably looked at piecewise smooth curves. The following is an attempt to generalize this to the present situation.
Definition 10.2.6 In the above context, I will call Ω a PC^{1} manifold if it is a C^{0} manifold with charts
 (10.4) 
Note that R_{i}
 (10.5) 
Also the following notation is often used with the convention that v = R_{i} ∘ R_{j}^{−1}

If you want to keep things more simple, just let L = ∅ and let it be a C^{1} manifold. This extra complication is just to allow creases in the surface. See the example below. To do this right, you would consider a Lipschitz manifold and use Rademacher’s theorem. However, the above allows for lots of pointy places and creases in the manifold. Note that the composition of Lipschitz maps is also Lipschitz.

Example 10.2.7 Let f : ℝ^{n+1} → ℝ and suppose lim_{}
Note that this includes S^{p−1} and lots of other things like x^{4} + y^{2} + z^{4} = 1 and so forth. The details are left as an exercise.
Proof: S_{j}
Note that on R_{j}

Allowing this to take place will allow for the possibility that Ω could have creases in it. For example, you could consider

When you graph this function letting x =