Definition 1.3.3 (Forms) . Given open U ⊆ R $^{n}$, a differential m -form on U is a function φ : U → ∧ $^{m}$( R $^{n}$) . The set of all m -forms on U is denoted by D $^{m}$( U ) . We say that φ ∈ D $^{m}$( U ) is C k if φ ( x ) applied to v$_{1}$ ∧ v$_{2}$ ∧ · · · ∧ v$_{m}$ is a C k function in x for any fixed vectors v$_{i}$ ∈ R $^{n}$. Observe that any function f : U → R can be considered as a 0-form. Differential m -forms can be used as integrands over m -dimensional surfaces as they can vary both based on location of a point and its tangent plane; this serves as a useful generalization of integration of 1-forms over a curve. Definition 1.3.4 (Exterior differentiation) . Suppose U ⊂ R n is open and f : U → $_{R}$is C $^{1}$. The exterior derivative of the 0-form f is the 1-form d f defined by The exterior derivative of the simple m -form φ = f d x$_{i}$$_{1}$ ∧ d x$_{i}$$_{2}$ ∧ · · · ∧ d x$_{i}$$_{m}$ (where the i$_{k}$ are integers from 1 to n ) is given by the ( m + 1) -form For all other C 1 m -forms, the definition is extended by linearity. Theorem 1.3.5 (Properties of exterior differentiation, [29] p. 163) . If φ and ψ are C 1 m -forms and θ is a C$_{1}$ ℓ -form, then we have: 6 d φ = d f ∧ d x$_{i}$$_{1}$ ∧ d x$_{i}$$_{2}$ ∧ · · · ∧ d x$_{i}$$_{m}$. d f = ∂f ∂x$_{1}$ d x$_{1}$ + ∂f ∂x$_{2}$ d x$_{2}$ + · · · + ∂f ∂x$_{n}$ d x$_{n}$. • d( φ + ψ ) = d φ + d ψ • d( φ ∧ θ ) = (d φ ) ∧ θ + ( − 1) $^{m}$φ ∧ (d θ ) • If φ is C $^{2}$, then dd φ = 0 .