Let v_{1}, ..., v_{m} be a basis for V. The exterior power of V =

Λ^{p}V = {Σa_{i_{1}}...a_{i_{p}}v_{i_{1}} ∧ ... ∧ v_{i_{p}} | a_{i_{1}}...a_{i_{p}} ∈ R, i_{1} < ... < i_{p}}.

I.e., Λ^{p}V is the vector space over R with basis consisting of the ( \begin{matrix}n \\ p \end{matrix} ) elements v_{i_{1}} ∧ ... ∧ v_{i_{p}}. Recall v ∧ v = 0 implies v_{i} ∧ v_{j} = v_{j} ∧ v_{i} Λ^{∗}V = ∪_{p=0}^{∞}Λ^{p}V is a graded associative algebra with addition + and multiplication ∧. If v ∈ Λ^{p}V then v has degree p (i.e., it’s in the pth grade). Λ^{0}V = R which is generated by 1, the multiplicative identity for this algebra. The wedge product is bilinear: (c_{1}v_{1} + c_{2}v_{2}) ∧ w = c_{1}(v_{1} ∧ w) + c_{2}(v_{2} ∧ w) and v ∧ (c_{1}w_{1} + c_{2}w_{2}) = c_{1}(v ∧ w_{1}) + c_{2}(v ∧ w_{2}) Prop 5.2: α ∈ Λ^{p}V, β ∈ Λ^{q}V, then α ∧ β = (−1)^{pq}β ∧ α. Prop 5.6: If A : V → W is a linear transformation, then Λ^{p}A : Λ^{p}V → Λ^{p}V defined by Λ^{p}A(v_{1} ∧...∧v_{p}) = Av_{1} ∧...∧Av_{p} is a linear transformation which is independent of the choice of basis vectors. Prop 5.7: If dimV = n, Λ^{n}A : Λ^{n}V → Λ^{n}V is given by Λ^{n}A(cv_{1} ∧ ... ∧ v_{n}) = (det A)(cv_{1} ∧ ... ∧ v_{n}). Defn: The dual of T_{p}M = T_{p}^{∗}M is the cotangent space to M at p. If \frac{∂}{∂x_{1}} , ..., \frac{∂}{∂x_{m}} is a basis for T_{p}M, then the dual basis will be denoted dx_{1}, ..., dx_{m}. Λ^{p}T_{x}^{∗}M is the p-fold exterior power of the vector space T_{x}^{∗}M The bundle of p-forms = Λ^{p}T^{∗}M = ∪_{x∈M}Λ^{p}T_{x}^{∗}M a manifold of dimension m +

( \begin{matrix}m \\ p \end{matrix} ) .

Let π : Λ^{p}T^{∗}M → M, π(x, v) = x. If (U, ϕ) is a chart for M, then the following is a chart for Λ^{p}T^{∗}M

Φ : π^{−1}(U) → ϕ(U) × Λ^{p}R^{m} ⊂ R^{m} × R^{( \begin{matrix}m \\ p \end{matrix} )}

Φ(x, v) = (ϕ(x), y_{1}, ..., y_{( \begin{matrix}m \\ p \end{matrix} )}) where y_{i} are the coefficients of v in terms of a given basis for Λ^{p}T_{x}^{∗}M A differential p-form is a section of Λ^{p}T^{∗}M I.e, a vector field = smooth function s : M → Λ^{p}T^{∗}M, s(x) = (x, v_{x}) Ex: 0-form = smooth function. 1-form: Σa_{i}(x)dx_{i}, locally defined. 6.2: Partitions of unity Suppose f : M → R. Defn: supp f = \overline{\{x ∈ M | f(x) ≠ 0\}} Defn: {D_{α}}_{α∈A} is locally finite if ∀x ∈ M, there exists U open such that U ∩D_{α} = ∅ for all but finitely many α. Defn: A partition of unity on M is a countable collection {f_{i}}_{i∈I} of smooth functions such that 1.) f_{i} ≥ 0 2.) {supp f_{i} | i ∈ I} is locally finite 3.) Σ_{i}f_{i} = 1 Thm 6.1: Given any open covering {V_{α}} of a manifold M, there exists a partition of unity {f_{i}} such that supp f_{i} ⊂ V_{α_{i}} for some α_{i}. Such a partition of unity is said to be subordinate to the covering {V_{α}}. Prop: Change of basis: dy_{i_{k}} = Σ\frac{∂y_{i_{k}}}{∂x_{j}} dx_{j} 6.4 The exterior derivative. Let Ω^{p}(M) = vector space of all p-forms on M d : Ω^{p}(M) → Ω^{p+1}(M) If f ∈ Ω^{0}(M) (i.e., f is smooth): df = Σ_{i=1}^{n} \frac{∂f}{∂x_{i}} dx_{i} = (\frac{∂f}{∂x_{1}} , ..., \frac{∂f}{∂x_{m}}) where basis = {dx_{1}, ..., dx_{n}} If α = Σ_{i=1}^{n}a_{i_{1}...i_{p}}(x)dx_{i_{1}} ∧ ... ∧ dx_{i_{p}} ∈ Ω^{p}(M), let dα = Σ_{i=1}^{n}da_{i_{1}...i_{p}}(x) ∧ dx_{i_{1}} ∧ ... ∧ dx_{i_{p}} ∈ Ω^{p}(M) Note d^{2}α = 0 and d(α ∧ β) = dα ∧ β + (−1)^{p}α ∧ dβ for α ∈ Ω^{p}(M) Ex: If α = a_{1}(x)dx_{1} + a_{2}(x)dx_{2} + a_{3}(x)dx_{3} dα = da_{1}(x)dx_{1} + da_{2}(x)dx_{2} + da_{3}(x)dx_{3} = [\frac{∂a_{1}}{∂x_{1}} dx_{1} + \frac{∂a_{1}}{∂x_{2}} dx_{2} + \frac{∂a_{1}}{∂x_{3}} dx_{3}] ∧ dx_{1} + [\frac{∂a_{2}}{∂x_{1}} dx_{1} + \frac{∂a_{2}}{∂x_{2}} dx_{2} + \frac{∂a_{2}}{∂x_{3}} dx_{3}] ∧ dx_{2} + [\frac{∂a_{3}}{∂x_{1}} dx_{1} + \frac{∂a_{3}}{∂x_{2}} dx_{2} + \frac{∂a_{3}}{∂x_{3}} dx_{3}] ∧ dx_{3} [0 + \frac{∂a_{1}}{∂x_{2}} dx_{2} ∧ dx_{1} + \frac{∂a_{1}}{∂x_{3}} dx_{3} ∧ dx_{1}] + [\frac{∂a_{2}}{∂x_{1}} dx_{1} ∧ dx_{2} + 0 + \frac{∂a_{2}}{∂x_{3}} dx_{3} ∧ dx_{2}] + [\frac{∂a_{3}}{∂x_{1}} dx_{1} ∧ dx_{3} + \frac{∂a_{3}}{∂x_{2}} dx_{2} ∧ dx_{3} + 0] [\frac{∂a_{3}}{∂x_{2}} − \frac{∂a_{2}}{∂x_{3}} ]dx_{2} ∧ dx_{3} + [\frac{∂a_{1}}{∂x_{3}} − \frac{∂a_{3}}{∂x_{1}} ]dx_{3} ∧ dx_{1} + [\frac{∂a_{2}}{∂x_{1}} − \frac{∂a_{1}}{∂x_{2}} ]dx_{1} ∧ dx_{2} (\frac{∂a_{3}}{∂x_{2}} − \frac{∂a_{2}}{∂x_{3}} , \frac{∂a_{1}}{∂x_{3}} − \frac{∂a_{3}}{∂x_{1}} , \frac{∂a_{2}}{∂x_{1}} − \frac{∂a_{1}}{∂x_{2}} ) = ∇ × (a_{1}, a_{2}, a_{3}) Note d^{2}α = 0 Ex: d([\frac{∂a_{3}}{∂x_{2}} − \frac{∂a_{2}}{∂x_{3}} ]dx_{2} ∧ dx_{3} + [\frac{∂a_{1}}{∂x_{3}} − \frac{∂a_{3}}{∂x_{1}} ]dx_{3} ∧ dx_{1} + [\frac{∂a_{2}}{∂x_{1}} − \frac{∂a_{1}}{∂x_{2}} ]dx_{1} ∧ dx_{2}) = d[\frac{∂a_{3}}{∂x_{2}} − \frac{∂a_{2}}{∂x_{3}} ]dx_{2} ∧ dx_{3} + d[\frac{∂a_{1}}{∂x_{3}} − \frac{∂a_{3}}{∂x_{1}} ]dx_{3} ∧ dx_{1} + d[\frac{∂a_{2}}{∂x_{1}} − \frac{∂a_{1}}{∂x_{2}} ]dx_{1} ∧ dx_{2}) = [Σ_{i=1}^{n} \frac{∂^{2}a_{3}}{∂x_{2}∂x_{i}} − \frac{∂^{2}a_{2}}{∂x_{3}∂x_{i}} dx_{i}] ∧ dx_{2} ∧ dx_{3} + [Σ_{i=1}^{n} \frac{∂^{2}a_{1}}{∂x_{3}∂x_{i}} − \frac{∂^{2}a_{3}}{∂x_{1}∂x_{i}} dx_{i}] ∧ dx_{3} ∧ dx_{1} + [Σ_{i=1}^{n} \frac{∂^{2}a_{2}}{∂x_{1}∂x_{i}} − \frac{∂^{2}a_{1}}{∂x_{2}∂x_{i}} dx_{i}] ∧ dx_{1} ∧ dx_{2} = 0