By Lisle I.G.

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**Example text**

The k-th order contact forms C on the space (x, u, u, . . 15) The tangency conditions require that C be preserved by a transformation τ . 3. The (formally inﬁnite) diﬀerential operator Dxi = ∂ ∂ ∂ + uji + · · · + ujIi + ··· . 16) is called the total derivative with respect to xi . Total derivative operators Dxi are naturally dual to contact forms C in the k sense that they are annihilated by every such form. Although the sum deﬁning Dxi is formally inﬁnite, we only apply total derivative operators to functions f (x, u, u, 1 .

Inserting this transformation τ into any equation E(φ) ∈ C maps it to another equation E(φ ) ∈ C. Most importantly, the relationship between the original arbitrary element a = φ(w) and its transform a = φ (w ) is the result of a transformation τˆ acting on (w, a) space as τˆ : w = τ (w) a = σ(w, a). 2. Equivalence transformations (Recall that w ≡ (x, u)). That is, φ (w ) = σ τ −1 (w ), φ ◦ τ −1 (w ) . ’s. ’s. 18) acting on (w, a) space will be called an augmented transformation. We follow the convention that an object (such as a transformation) on augmented space is ‘hatted’ (ˆ τ ); its projection to a corresponding object on base space is unhatted (τ ).

1 In contrast, the components of a are values 1 of derivatives of a = φ(w) with respect to w, so a = {aβγ }, for β = 1, . . , µ and γ = 1, . . , ν, where aβγ = ∂φβ ∂wγ (w). 1 are a = (ax , at , au ), whereas u = (ux , ut ). 2. 11) where c(x) is an arbitrary wavespeed function representing spatial inhomogeneity of the medium. 12) and primary system E: utt = a2 uxx . 11) results when we assign a = c(x) as the general solution of A. Potential forms of wave equations can be constructed. 11). 11).

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