This seventh edition of Advanced Engineering Mathematics differs from the sixth in four ways. First, based on reviews and user comments, new material has been added, including the following. ? Orthogonal projections and least squares approximations of vectors and functions. This provides a unifying theme in recognizing partial sums of eigenfunction expansions as projections onto subspaces, as well as understanding lines of best fit to data points. ? Orthogonalization and the production of orthogonal bases. ? LU factorization of matrices. ? Linear transformations and matrix representations. ? Application of the Laplace transform to the solution of Bessel’s equation and to problems involving wave motion and diffusion. ? Expanded treatment of properties and applications of Legendre polynomials and Bessel functions, including a solution of Kepler’s problem and a model of alternating current flow. ? Heaviside’s formula for the computation of inverse Laplace transforms. ? A complex integral formula for the inverse Laplace transform, including an application to heat diffusion in a slab. ? Vector operations in orthogonal curvilinear coordinates. ? Application of vector integral theorems to the development of Maxwell’s equations. ? An application of the Laplace transform convolution to a replacement scheduling problem.