  A thorough discussion and development of the calculus of real-valued
functions of complex-valued vectors is given using the framework of the
Wirtinger Calculus. The presented material is suitable for exposition in an
introductory Electrical Engineering graduate level course on the use of complex
gradients and complex Hessian matrices, and has been successfully used in
teaching at UC San Diego. Going beyond the commonly encountered treatments of
the first-order complex vector calculus, second-order considerations are
examined in some detail filling a gap in the pedagogic literature.
