Chapter 6 Partial Differential Equations

6.2. LINEAR AND QUASILINEAR EQUATIONS OF FIRST ORDER 9
also be represented in parametric form as the image of an open set in R n−1 (with coordinates
x ′ ) under the map
x ′ ↦→ (x ′ ,ψ(x ′ )).
Thus x ′ can be thought of as giving local coordinates on S near x 0 .
A hypersurface S is called characteristic for L at x if the normal vector ν(x) is in Char x (L)
and S is called non-characteristic if it is not characteristic at any point.
An important property of the characteristic variety is contained in the following:
Let F be a smooth one-to-one mapping of Ω onto Ω ′ ⊂ R n and set y = F (x).
Assume that the Jacobian matrix
[ ] ∂yi
J x = (x)
∂x j
is nonsingular for x ∈ Ω, so that {y 1 ,y 2 , ··· ,y n } is a coordinate system on Ω ′ .
We have
∂
n∑ ∂y i ∂
=
∂x j ∂x j ∂y i
i=1
which we can write symbolically as ∂ x = Jx T ∂ y , where Jx
T is the transpose of
J x . The operator L is then transformed into
L ′ = ∑ (
a α F −1 (y) ) ( α
J T F −1 (y) y) ∂ on Ω ′ .
|α|≤k
When this expression is expanded out, there will be some differentiations of J T F −1 (y) but
such derivatives are only formed by “using up” some of the ∂ y on J T F −1 (y), so they do not
enter in the computation of the principal symbol in the y coordinates, i.e., they do not enter
the highest order terms. We find that
χ L (x, ξ) = ∑
( ) α
a α (F −1 (y)) J T F −1 (y) ξ .
|α|=k
Now since F −1 (y) =x, on comparing with the expression
χ L (x, ξ) = ∑
a α (x)ξ α
|α|=k
we see that Char x (L) is the image of Char y (L ′ ) under the linear map J T F −1 (y) .