§2 Formulae

• Old independent coordinates
• New independent coordinates

(It may include time as one coordinate). Assume

The original n-dimensional second order quasilinear equation:

where is a symmetric matrix and .

To convert:

• The new matrix coefficients are

  

• The new lower order terms are

  

• In the coefficients, express in terms of by solving the given relationships for them;
• In the coefficients, rewrite the first order derivatives:

  

Matrix notation: Let B^T be the matrix of new coordinate derivatives:

then

A' = B^T A B

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