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Section 6.3 Differential Forms

Given a vector space \(V\) with nondegenerate inner product \(g\text{,}\) we can also construct the exterior algebra \(\bigwedge(V)\text{.}\) As in Section 6.2, we extend \(V\) to an algebra by introducing a bilinear, associative product. However, this time the product, written as \(\wedge\) (“wedge”), is antisymmetric, that is

\begin{equation} w\wedge v = -v\wedge w\tag{6.3.1} \end{equation}

for all \(v,w\in V\text{,}\) or equivalently

\begin{equation} v\wedge v = 0\tag{6.3.2} \end{equation}

for all \(v\in V\text{.}\) Multiplying \(k\) elements of \(V\) together yields an element of rank \(k\text{,}\) which will only be nonzero if the elements are linearly independent. Thus, as before, a basis \(\{e_m\}\) of \(V\) extends to a basis

\begin{equation} e_{m_1...m_k} = e_{m_1} \wedge ... \wedge e_{m_k}\tag{6.3.3} \end{equation}

of \(\bigwedge(V)\text{.}\) We include the scalars (normally \(\RR\)) as elements of rank \(0\text{,}\) with basis element \(1\text{,}\) by defining

\begin{equation} 1\wedge v = v .\tag{6.3.4} \end{equation}

There are thus \({n\choose k}=\frac{n!}{k!(n-k)!}\) independent elements of rank \(k\text{,}\) and \(n^2\) total elements, in \(\bigwedge(V)\text{.}\)

This entire construction extends easily to the case of vector fields, where the scalars are functions. A common application is to consider vector fields on a manifold \(M\text{,}\) that is, elements of the tangent bundle. In this case, one normally uses the dual space \(TM^*\) of linear operators on \(TM\text{.}\)

An example of such a dual vector field is the differential \(df\) of a smooth function \(f\text{.}\) How does it act on a vector field \(X\text{?}\) By taking the directional derivative of \(f\) in the direction of \(X\text{,}\) that is,

\begin{equation} df(X) = X(f) = X^m e_m(f) .\tag{6.3.5} \end{equation}

If \(\{e_m\}\) is a coordinate basis, then it acts on functions by partial differentiation, that is,

\begin{equation} e_m(f) = \frac{\partial f}{\partial x^m} ,\tag{6.3.6} \end{equation}

but in general the action of \(e_m\) is a linear combination of such derivatives.

Similarly, the general dual vector (field) is a linear combination of such differentials. These dual vectors are called 1-forms, and their products are called k-forms, with functions being 0-forms by definition. Elements of the resulting exterior algebra \(\bigwedge(M)\) are called differential forms. If \(\{e_m\}\) is a basis for vector fields, the dual basis \(\sigma^m\) of \(V^*\) is defined by

\begin{equation} \sigma^i(e_j) = \delta^i{}_j\tag{6.3.7} \end{equation}

where \(\delta^i{}_j\) denotes the Kronecker delta.