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Unformatted text preview: Math 425, Spring 2011 Jerry L. Kazdan PDE: Linear Change of Variable Let x : = ( x 1 , x 2 ,..., x n ) be a point in R n and consider the second order linear partial differential operator Lu : = n ∑ i , j = 1 a i j ∂ 2 u ∂ x i ∂ x j , (1) where the coefficient matrix A : = ( a i j ) is constant. Since for functions whose second derivatives are con- tinuous we know that ∂ ∂ x i ∂ ∂ x j = ∂ ∂ x j ∂ ∂ x i we may (and will) assume that A is a symmetric ma- trix: A = A * . In these brief notes we obtain a useful formula for how L changes if we make the linear change of vari- able y = Sx where ( S : = s k ‘ ) is a constant matrix. Written in coordinates this means that y k = n ∑ ‘ = 1 s k ‘ x ‘ , where k = 1 ,..., n . F IRST G OAL : Compute L in these new y coordi- nates. This is straightforward (even boring) if you 1 just keep calm and don’t make copying errors. By the chain rule ∂ u ∂ x j = n ∑ k = 1 ∂ u ∂ y k ∂ y k ∂ x i = n ∑ k = 1 ∂ u...
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