DifferentialForm

A differential form is a concept used in calculus, specifically in the field of differential geometry and multivariable calculus. It provides a unified approach to define integrals of functions over curves, surfaces, volumes, and higher-dimensional manifolds.

Before delving into the details, let's set up some notations and conventions. Suppose we're working in a smooth manifold $M$ of dimension $n$, and $p \in M$ is a point.

1. Tangent Space: The tangent space at $p$, denoted $T_pM$, is the vector space of all tangent vectors at $p$. The dimension of $T_pM$ is the same as that of $M$.

2. Cotangent Space: The cotangent space at $p$, denoted $T^*_pM$, is the dual space of $T_pM$. It consists of all linear maps from $T_pM$ to the real numbers $\mathbb{R}$. It is also a vector space of the same dimension as $M$.

A differential form on $M$ is a smooth section of the cotangent bundle, which is the collection of all cotangent spaces at every point in $M$.

Specifically, a differential 1-form is a smooth function that assigns to every point $p \in M$ a linear functional $\alpha_p: T_pM \rightarrow \mathbb{R}$. This means for each $p$, $\alpha_p$ takes as input a tangent vector at $p$ and outputs a real number.

In coordinates $(x^1, ..., x^n)$ on $M$, a differential 1-form $\omega$ can be written as:

$$\omega = \Sigma a_i dx^i$$ (sum from $i=1$ to $n$)

where $a_i$ are smooth functions on $M$, and $dx^i$ are the coordinate 1-forms, defined as $dx^i ( \partial/\partial x^j ) = \delta^i_j$.

A differential $k$-form is a smooth section of the $k$-th exterior power of the cotangent bundle. This means for every point $p \in M$, a differential $k$-form assigns a multilinear, skew-symmetric, real-valued function on the $k$-tuples of tangent vectors at $p$.

In coordinates, a differential $k$-form $\omega$ can be written as:

$$\omega = \Sigma a_I dx^I$$ (sum over multi-index $I$)

where $a_I$ are smooth functions and $dx^I = dx^{i1} \wedge ... \wedge dx^{ik}$ is the exterior product of 1-forms. The symbol "$\wedge$" denotes the wedge product, which is a special operation in the algebra of differential forms that generalizes the cross product.

The set of all $k$-forms on $M$ forms a vector space, often denoted $\Omega^k(M)$.

The exterior derivative is an operation $d: \Omega^k(M) \rightarrow \Omega^{k+1}(M)$ that takes a $k$-form and produces a $(k+1)$-form. The operation $d$ satisfies three properties:

1. Linearity:

$$d(\alpha + \beta) = d\alpha + d\beta$$

and

$$d(f\alpha) = df \wedge \alpha + f d\alpha \forall k-\text{ forms } \alpha, \beta \text{ and smooth functions } f$$

2. Leibniz rule (product rule):

$$d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta$$ if $\alpha$ is a $k$-form.

3. $d^2 = 0$: applying $d$ twice gives the zero map.

Differential forms, along with the exterior derivative, form the basis for de Rham cohomology, a topological invariant of manifolds. Integral calculus on manifolds, including the fundamental theorem of calculus, Stokes' theorem, and others, can be formulated in terms of differential forms.