The question I ask is in the title. This should be quite well-known, and in fact probably I am going to get the response that it is the definition. To convey my confusion, I have to convey my understanding of what is a differential form and what is the contangent bundle. To simplify things, we assume that our whole setup is immersed in the Euclidean space.

$1$. **Differential forms**.

We take the definitions from Rudin, Principles of Mathematical Analysis. This is for an open set in $\mathbb R^n$.

Suppose $E$ is an an open set in $\mathbb R^n$. A $k$-surface in $E$ is a differentiable mapping $\Phi$ from a compact subset $D \subset \mathbb R^k$ into $E$. $D$ is called the parameter domain of $\Phi$ consisting of points $\mathbf u = ( u_{i_1}, \cdots , u_{i_k} )$.

A differential form of order $k \geq 1$ in $E$ is a function $\omega$, symbolically represented by the sum

$$\omega = \sum a_{i_1, \cdots , i_k}(\mathbf x) dx_{i_1}\wedge \cdots \wedge dx_{i_k}$$

where the indices $i_1, \cdots , i_k$ range independently from $1$ to $n$, and so that $\omega$ assigns to each $k$-surface $\Phi$ in $E$ a number$\omega(\Phi) = \int_\Phi \omega$ , according to the rule

$$\int_\Phi \omega = \int_D \sum a_{i_1, \cdots , i_k}(\Phi((\mathbf{u})) \frac{\partial ( x_{i_1}, \cdots , x_{i_k})}{\partial ( u_{i_1}, \cdots , u_{i_k})}d\mathbf u $$

where $D$ is the parameter domain of $\Phi$, and the functions $a_{i_1}, \cdots, a_{i_k}$ are assumed to be real and continuous in $D$.

So in the above definition the differential $k$-form is a certain integral for functions on compact $k$-surfaces. Thus a differential form can be treated as a measure for the $k$-surfaces, which can be integrated.

$2$. **Cotangent bundle**

We take this from wikipedia.

Let $M \times M$ be the Cartesian product of $M$ with itself. The diagonal mapping $\Delta$ sends a point $p$ in $M$ to the point $(p,p)$ of $M \times M$. The image of $\Delta$ is called the diagonal. Let $\mathcal{I}$ be the sheaf of germs of smooth functions on $M \times M$ which vanish on the diagonal. Then the quotient sheaf $\mathcal{I}/\mathcal{I}^2$ consists of equivalence classes of functions which vanish on the diagonal modulo higher order terms. The cotangent sheaf $\Omega$ is the pullback of this sheaf to $M$.

Now, Def 2: A differential form $k$-form $\omega$ is a section of $\wedge^k\ \Omega$.

**Question**.

We consider an open set in the Euclidean space and look at the two definitions. A priori, to my eyes, both appear to be different things. It needs to be proved that they are the same. Please help me out with a reference with the required proofs.