Mathematical Methods in the Physical Sciences · Problem 9.13

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · conformal-mapping-matrix

Problem

Verify the matrix equation (dudv)=(J)(dxdy),\begin{pmatrix} du \\ dv \end{pmatrix} = (J) \begin{pmatrix} dx \\ dy \end{pmatrix}, where (J)(J) is a matrix whose determinant is the Jacobian in Problem 12. (Also see Chapter 10, Section 13.) Multiply the matrix equation by its transpose and use Problem 12 to obtain dS/ds=∣dw/dz∣dS/ds = |dw/dz| as in (9.7).

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